on fibonacci sequence and its extensions
TRANSCRIPT
ON FIBONACCI SEQUENCE AND ITS EXTENSIONS
THESIS SUBMITTED FOR THE AWARD OF THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN THE FACULTY OF NATURAL SCIENCES
GOA UNIVERSITY5 I K 5 _
P h A / F f b
BY
CH ANDRAKANT NAGNATH PHADTE
DEPARTMENT OF MATHEMATICS
GOA UNIVERSITY
814
DECLARATION
I, the undersigned, hereby declare that the thesis titled "On Fibonacci Se
quence and Its Extensions" has been completed by me and has not pre
viously formed the basis for the award of any diploma, degree, or any other
similar titles.
Chandrakant N Phadte
Taleigao Plateau-Goa
24-06-2016
/\\\ Sw^e,S VlcI Cor^tck<,KS “We. CV'A S H C - V
I'LsA e\y-«cv «-* y-wi .
CERTIFICATE
This is to certify that Mr. Chandrakant Nagnath Phadte has successfully com
pleted the thesis entitled "On Fibonacci Sequence and Its Extensions"
for the degree of Doctor of Philosophy in Mathematics under my guidance
during the period 2012-2016 and to the best of my knowledge it has not pre
viously formed the basis of award of any degree or diploma in Goa University
or elsewhere.
Guide,
Associate Professor
Department of Mathematics
Heaa7x)epartment of Mathematics
Goa University
ACKNOWLEDGEMENT
I wish to express my most sincere gratitude to Dr. S.P. Pethe, former vis
iting Professor, Department of Mathematics, Goa University. He suggested
the problem under taken for this study. The guidance, support and immense
mathematical knowledge that I received from Dr. S. P. Pethe while working
on my thesis has been greatly appreciated. I would like to thank Dr. Y. S.
Valaulikar for his guidance and friendship from the first day I arrived in the
department. I am especially grateful for his time and effort. I am very much
thankful to Principal P. M. Bhende for the inspiration and encouragement
given to me. My thanks are also due to Dr. M. Tamba, Lecturer, Department
of Mathematics, Goa University. I would also like to thank the faculty and
the non-teaching staff of the Department of Mathematics, Goa University for
their selfless service.
Lastly I would also like to thank my family who always helped me when I
needed it.
Contents
1 Introduction 1
2 Literature Review 4
2.1 Introduction.................................................................................... 4
2.2 Binet Formula................................................................................. 7
2.3 Plethora of Fibonacci Identities:................................................... 9
2.4 Generating F unc tion ........................................................................ 11
2.5 Various Extensions of Fibonacci Sequence .....................................12
2.5.1 Horadam’s Extension............................................................ 12
2.5.2 Fibonacci Polynomials......................................................... 15
2.5.3 Fibonacci Function by Francis Parker..................................15
2.5.4 Extension due to Horadam and Shannon .........................16
2.5.5 Elmore’s Extension............................................................... 16
2.5.6 Extension by J.E.Walton and A.F. H o ra d a m ...................17
2.5.7 Extension to Complex Fibonacci N um bers........................ 18
2.5.8 Extension to Tribonacci Numbers....................................... 20
2.5.9 Extension of Fibonacci sequence using Generalized Cir
cular Functions.....................................................................20
2.6 Fibonacci Numbers and Binomial Coefficients.............................. 24
2.7 Divisibility properties of Fibonacci N um bers.................................29
1
3 Pseudo Fibonacci Sequence 31
3.1 Introduction...................................................................................... 31
3.2 Some Fundamental Identities of {gn} ............................................. 32
3.3 A Generalization to a new Sequence {Gn} ....................................40
3.4 Some Fundamental Identities of Gn ................................................ 43
3.5 E x am p les..........................................................................................59
3.6 Properties of {Gn} using Matrices ................................................ 61
3.7 Another Generalization..................................................................... 67
3.8 Use of Generalized Circular Functions............................................. 69
3.9 Some Identities of Hn( x ) .................................................................. 72
4 Pseudo Tribonacci Sequence 78
4.1 Introduction.......................................................................................78
4.2 Some identities of {Jn} .....................................................................81
4.3 Use of E-Operator ...........................................................................91
4.4 Generalization of {Jn} .....................................................................93
5 Pseudo Fibonacci Polynomials 95
5.1 Introduction...................................................................................... 95
5.2 Some Fundamental Identities of gn( x , t ) .......................................... 99
5.3 Pseudo Fibonacci Polynomial in two variables............................ 105
6 Congruence Properties of Gn 110
6.1 Introduction....................................................................................110
2
6.2 Some Identities 112
6.3 Modular Properties.........................................................................114
SUMMARY 119
LIST OF PUBLICATIONS/COMMUNICATIONS 119
BIBLIOGRAPHY 119
3
Chapter 1
Introduction
Fibonacci Sequence have intrigued Mathematicians for years. Generalized Fi
bonacci Sequence can be noticed in many fields like computer algorithms,
cryptography, optical network, probability theory and so on. There are many
studies in literature that are concerned about the general sequences of second
order. For example the Lucas sequence, Jacobsthal sequence, k - Fibonacci
sequence, etc. [24], [20],[4].
The main objective of this research is to study the well known Fibonacci se
quence and its identities with the intention of generalizing the results to a gen
eral sequence defined by second order non-homogeneous recurrence relation. It
is well known that the Fibonacci retracement play an important part in stock
trading. It is natural to expect that the non homogeneous recurrence relation
associated with Fibonacci relation will also be useful in the stock analysis es
pecial when the non homogeneous term is oscillatory. With this application
in mind the non homogeneous relation of the type gn+2 = gn+1 + 9n + Atn
was formed, where A is a non zero constant and t is some fixed constant. The
sequence generated by this relation is named as pseudo Fibonacci sequence
indicating that the new sequence is like Fibonacci sequence and hence should
1
exhibit the properties /identities in line with those of Fibonacci sequence. The
main part of the research involves in exploring the generalizations of Fibonacci
Sequence in this direction and obtaining a more generalized sequence together
with its different identities. The richness of the results in generalization work
prompted our investigation on this topic.
In this thesis an attempt is made to develop the theory of generalizations of
Fibonacci sequence by introducing non homogeneous term and also by chang
ing the seed values. The thesis is divided into six chapters with Chapter 1
introducing the topic and describing the thesis. Chapter 2 opens with rabbit
problem. After defining the Fibonacci sequence, it deals with a survey of var
ious important properties and main generalizations of this sequence. In short,
this chapter is a survey of existing literature on Fibonacci sequence and its
extensions.
In Chapter 3, we have introduced a new generalization {<?„}, of the Fibonacci
sequence, defined by non homogeneous recurrence relation and called it pseudo
Fibonacci sequence. This sequence is further extended to obtain another gen
eralization {Gn} • The various identities of these sequences are stated and
proved in section (3.4), (3.5). In section (3.10), further generalization of {G„}
with its properties are discussed. Few results are verified by means of examples
in section(3.11).
In chapter 4, the third order non-homogeneous recurrence relation has been
studied to extend our concepts to Tribonacci sequence.
2
In chapter 5, another extension of Fibonacci sequence that gives rise to a new
class of Fibonacci polynomials has been discussed. Some identities of these
newly obtained pseudo Fibonacci polynomials are proposed and proved.
Chapter 6 contains pseudo Fibonacci sequence modulo m, a positive integer
with the intentions of generalizing result to sequence obtained from pseudo
Fibonacci sequence. This mainly involves in investigating the periodicity of
the new sequence after modding out by m.
Finally, a brief summary of the work done is presented together with the future
plans. The thesis ends with a Bibliography.
3
Chapter 2
Literature Review
2.1 Introduction
The Fibonacci sequence of numbers is named after its promoter the Italian
mathematician, Leonardo Pisano also known as Fibonacci. In his book “Liber
Abacci” meaning book of calculation or book of counting, published in 1202, he
discussed the problem of rabbit regeneration. In relation with the generation
of rabbits he posed the following problem. Suppose each month the female
of a pair of rabbits gives birth to a pair of rabbits (of different sexes). Two
months later the female of new pair gives birth to a pair of rabbits. What is
the number of pairs of rabbits at the end of the year if there was one pair of
rabbit in the beginning of the year?.
At the end of the first month there will be 2 pairs of rabbits. At the end of
second month just one of these two pairs will have offspring and so the number
of pairs of rabbits will be 3. At the end of the third month the original pair
of rabbit as well as the pair born at the end of first month will have offspring
and so the number of pairs of rabbits will be 5.
Let Fn be the number of pairs of rabbits at the end of nth month. At the end
4
of (n + l) th month there will be Fn pairs of old rabbits and as many pairs of
new rabbits as there were pairs of rabbits at the end of (n — l)th month, that
is F„_i. Mathematically this can be written as
■ n+1 — Fn + Fn- 1.
As we have F\ — 1, F2 = 1, it follows that F3 = 2, F4 = 3, F5 = 5 and so on.
In particular, at the end of one year the number of rabbits equals to Fi2 — 233.
If the total number of rabbits for different months is listed one after other, it
gives rise to a sequence of numbers as
1,1,2,3,5,8,13,21,34,55,...
This sequence of numbers is known as Fibonacci Sequence. Here every term
of the sequence is obtained by adding the preceding two consecutive numbers.
We take one more problem which is structurally similar to Fibonacci’s rabbit
problem [Tucker, 1980,pp. 112-113].
Consider a staircase having n steps. One can climb it by taking one step or
two steps to begin with. How many ways can be there to climb the staircase?
Let us say there are Sn ways to climb the staircase. When one starts to climb,
he takes one step or two to begin with. If he takes one step then there are
Sn_ i steps to continue climbing the remaining n — 1 steps. If he takes two
steps then there remains Sn- 2 ways to climb the steps.
Hence,
Sn = S, 1 + <Sn-2
5
are the possibilities to continue climbing the remaining n — 1 steps. This
relation is equivalent to above mentioned equation. Here Si — 1 and S2 — 2.
This is again a Fibonacci sequence shifted by one term.
S n — F n + l-
In India, the discovery of these numbers was done much earlier in the 6th
century, in connection with Sanskrit Prosody. Much of the emphasis was laid
to study the effect in mixing the long (L) syllables with the short (S), giving
different patterns of L and S within a given fixed length resulting in Fibonacci
numbers. Paramanand Sing [16] mentions that Acharya Pingala (possibly
500 BC) was the first Mathematician to know these numbers. It is said that
Acharya Virahanka (6th century AD) was the first Indian Mathematician to
give a written representation of so called Fibonacci numbers between 600 to
800 A.D. The search of relation of these numbers was continued in Indian po
etry even after Acharya Hemachandra [1088 -1173 ].
Presently in most of the cases Fibonacci sequence is defined as follows:
Definition The Fibonacci Sequence {Fn} is defined by the recurrence rela
tion
Fn = Fn_1+F„_2, n > 2 (2.1)
with initial values (or seed values)
F0 = 0 and Fx = 1. (2.2)
6
2.2 B in et Formula
Any term of {.F„} can be given by the recurrence relation (2.1). It will be a
tedious task to calculate Fn for large value of n. However, it would be easy
to calculate nth Fibonacci number directly if one knows the formula for Fn.
In 1843, Jacques Philippe - Marie Binet designed a formula for nth term of
the sequence. Binet formula provides a method for computing any Fibonacci
number Fn in terms of its index n without listing the previous (n — l)th terms
of the sequence. Let a and ft be the roots of x2 — x — 1 = 0. Then by the
standard linear difference equation method, the solution of (2.1) is given by
Fn = Clan + c2£n, (2.3)
a — (3
Substituting c\ and C2 in (2.3), we get
f3 — a
Fn =<xn - p n a - 0 ’
(2.4)
where ci and c2 are to be obtained from initial conditions (2.2). Thus we getf
C\ + c2 = 0
era + c2/d = 1.
On solving above equations we get
1 , 1C\ — ------- and c2 =
(2.5)
which is called Binet formula.
Note that a = 1 +0 5 and 0 - — -— are the roots of x2 - x - 1 = 0
7
corresponding to the recurrence relation (2.1).
Also
a + /3 = l, a — (3 = V5, a(3 = —1. (2.6)
Binet Formula can be used to find the sum of many series connected with
Fibonacci numbers. One can illustrate this with the following example:
To find the sum of series
F3 + ^6 + Fg + ... + F3n,
we have
i*3 + FIs + ... + F 3na 3 - (3Z a 6 - p a3n - (33n
v/5 + V5 y/S
= -\={a3 + a 6 + ... + cc3n - (33 - /36 - ... - /?3n] v5
1 a 3n+3 — a 3 /33n+3- /3 3 “ v/5l a 3 - 1 £3 - 1
Since
a 3 — l = a + a:2 — l = a + a + l = 2a,
similarly,
P3 - l = 2(3.
Hence,
F3 + Fe + ... + F3n — ^ = [—,3n+3 a /33n+3 _
2a 2 (3
V5
a 3n+2 _ Q2 _ 03n+2 + £2
8
_ 1 3n+2 _ £ 3„+2 a2_£2_ 2 V5 x/5
— g [- 3n+2 - M2]
F'An+2 ~ 1
2
Binet Formula has been used to prove different identities in chapters 2 and 3
ahead.
2.3 P lethora o f Fibonacci Identities:
The sequence of Fibonacci numbers possesses a number of interesting and
important properties. The following are some simple properties of {F„}.
a) The sum of the first n Fibonacci numbers is the Fibonacci numbers two
further of n minus 1.
± F k = Fn+2 - l .fc=0
b) The total of consecutive even positioned Fibonacci numbers is equal to the
Fibonacci number one further along the sequence minus 1.
y Fzk = Fzn+i — i- *:=0
c) The total of consecutive odd positioned Fibonacci numbers is equal to the
Fibonacci number that follows the last odd number in the sum.
y Fzk—i = Fin-k=1
9
d) The total of the squares of first n Fibonacci numbers is the product of last
number and the next Fibonacci number in the sequence.
tfe ~ FnFn+1,fc=0
e) The sum of the products of consecutive Fibonacci numbers is either the
square of a Fibonacci number or Square of a Fibonacci number m in u s 1.
n+i F%+1, when n is odd,
k=2F„+i — 1, when n is even.
*
f) The sum of squares of two consecutive Fibonacci numbers is equal to the
Fibonacci number in the sequence whose position number is the sum of their
position numbers.
F„ + Fl +1 = F2n+1-
g) For any four consecutive terms from Fibonacci sequence, difference of squares
of the two middle terms is the product of two outer terms.
n + l F l = Fn-iFrn+2-
h) The product of two alternative terms of Fibonacci sequence is the square
of the middle term between them plus 1 or minus 1.
Fn-iFn+iF% + 1, if n is even,
F% — 1, if n is odd.
i) The difference of the squares of two alternate Fibonacci numbers is the
Fibonacci number in the sequence whose position number is the sum of their
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position numbers.
j) A Fibonacci number Fmn is divisible by a Fibonacci number Fm.
In other words if p is divisible by q then Fp is divisible by Fq where m, n, p and
q are positive integers.
k) The sum of any ten consecutive Fibonacci numbers is divisible by 11.
l) Fn+m = FnFm+i -h Fn—iFm, Tn, n > 1.
All the above properties can be easily proved by mathematical induction.
2.4 G enerating Function
Generating function for the sequence a0,a i,a 2, ... is the function whose power
series representation an is the coefficient of xn. Generating function establishes
the connectivity between function of real variable and sequence of numbers.
They are often expressed in closed form by some expression.
The generating function for the Fibonacci sequence is given by
R.T. Harsen [5] has generalized this result by the relation
Fm 4" Fm—\X 1 — x — X2
11
OOE{x) = Y ,F nxn/ n!,
n=0
which can be expressed in terms of a, f3 as
p a x _ p fix
2.5 Various E xtensions of Fibonacci Sequence
Many have extended Fibonacci sequence in different ways. In doing so, some
of them have changed the initial values where as others altered the recurrence
relation. Here we give some of the extensions of {Fn}.
2.5.1 Horadam’s Extension
One of the most widely used extension of Fibonacci that was given by A.F.
Horadam [8], [7]. He defined the extended Fibonacci Sequence as follows:
Let {W„} be a sequence defined by
W„ = Wn(a, b,p, q) = pWn- 1 - qWn, n > 2
where p and q are arbitrary integers with Wo = a and W\ — b.
We observe that {W„} reduces to the Fibonacci sequence {Fn} when p — 1,
q — —l and a = 0, b = 1. i.e.,
Fn = Wn( 0,1, 1,-1).
Another form of generating function for Fibonacci sequence is Exponen
tial G enerating function given by
12
It is very interesting to see that Wn(a,b,p, q) itself can be modified to give
other forms of sequences. Some of its special cases can be seen in the following
table:
a b P q
1 Integers 0 1 2 l
2 Odd numbers 1 3 2 l
4 Geometric Progression 1 r r+1 r
6 Fermat’s Sequence 1 3 3 2
7 Fermat’s Sequence 2 3 3 2
8 Lucas Sequence 2 1 1 -1
Table A: Special cases of Horadam’s Sequence
Binet Formula for Wn is given by
(2.9)
here
/ — 2(6 — a/3), m = 2(6 — aa), d = a — /3 (2.10)
where a, /3 are the distinct roots of x2 — px + q = 0.
Note that a + (3 = p and a/3 = q.
Remark: For p = 1, q = -1 and a = 0,6 = 1 equation (2.10) reduces to
Wn = Fn =an — /?" a - /? ’
where a and /? are the roots o f x 2 — a: — 1 = 0.
P roperties o f Wn(a, b;p,q):
13
Wn(a, b;p, q) has very interesting properties which can be easily proved by us
ing Binet Formula. Some of its properties are listed below:
1 Z W k = Wn+2 ~ b ~ (p ~ 1)(Wn+1 - a) k=o p — q — 1 ’
*
o ^ xxr (1 + ? )(^ 2n+2 ~ o) - (Pq)(W2n+1 - W .X)I , yy2k — ------------------ 5— ——rr5------------------,k=o p2 - (q + l)2
o ^ xxt p(w 2n+2 - a) - q(l + q){W2n+i - W -1)' ' 2fc+l o / i i\2 *k=o p2 - (q + l)2
where IFlj = pa
4. E ( - i mk=0ku, _ (P + l) [ ( - l )nWn+i + a] - b + (-1 )n+lWn+2
p - q - l
5. E ( - l W n' kWk+1Wk = ^ ^ — ^ ,k=o P
6. W„+r = Wrun - qWr-iUn-i = WnUr - qWn- XUr. u
where Un = Wn(0,1 \p,q).
7. W 2 = Wn+xWn. x - e q n~l
where e — q[bW-X — a2].
Theorem 1. The generating function for {W„} is given by
W(x) =a + {b — ap)x 1 — px — qx2
(2.11)
Note: For a = 0,6 = 1 and p = l,q = -1 , equation (2.11) reduces to
generating function of {F„} as given in (2.7).
14
The exponential generating Wn is given by
E (x ) ^ [ leax - me0x}. (2.12)
Note: For p = 1, q — —1, o = 0, b = 1 and l = 2, m = 2, above reduces to
which is the exponential generating function for Fn as derived in the section
(2.4).
2.5.2 Fibonacci Polynomials
One of the generalizations of Fibonacci numbers is Fibonacci polynomials.
These polynomials are defined by the recurrence relation similar to Fibonacci
numbers. Fibonacci Polynomial is defined as follows:
Definition: The nth Fibonacci Polynomial Fn(x) is defined by the relation
Fn(x) = xF(n-\){x) + F(n_2){x) with F0(x) = 0, Fi(x) = 1.
Fibonacci polynomial reduces to Fibonacci numbers when x — 1. i.e., F„(l) =
Fn where Fn is the Fibonacci number.
2.5.3 Fibonacci Function by Francis Parker
Francis Parker [15] studied another form of Fibonacci polynomial called Fi
bonacci function and is defined by
F(x) = a cos(nir)a x~7E
(2.13)
15
Here a is the larger of the two roots. Observe that for x — n,
F(n) = an — cos(rar)a n
an _ (_l)»(_l)-»0»V5
an - /3 n V5 •
Thus F(x) reduces to Fn when x — n.
2.5.4 Extension due to Horadam and Shannon
Horadam and Shannon [10] extended (2.14) by defining the Fibonacci curve
as follows:fyX _ ®
F(x) = ---------j=------. Here F{x) reduces to Fn when x — n.V5
2.5.5 Elmore’s Extension
Elmore [3] used the concept of derivatives of function to extend the Fibonacci
sequence as follows:gdX _
Let F0(x) = ----- j=— , where a and ft are the roots of x2 — x — 1 = 0.V5
Define successively F\(x), F2(x), Fs(x). . . by
Fi(x) = Fq(x) =
F2(x ) = Fq (x ) =
aeax - Pe13*7E ’
a2eax — P2ePx
In general,/ nmeax — BmePx
Fm(x) = Fim)(x) = - ----- ^ ----- , m > 1.
16
We observe that Fm+1(x) - Fm(x) + Fm_1(x), when x = 0, Fm(0) - Fm. Thus
Fn(x) is another extension of Fn.
Taking a, /? as the roots of x2 — px + q = 0 and using similar process we define, a n e ™ - p n e f!x
Fn(x) = -----------------, where d — a - /3.
We easily see that F*(x) - pF*_x(x) - qF*_2(x). In this case F*(0) = F*
where F0* = 0, F* = 1, but F* = pF*_x - qF,n -2 -
2.5.6 Extension by J.E.Walton and A.F. Horadam
J.E.Walton and A.F. Horadam [28] generalized Fibonacci function by using
Elmore’s concept as follows:
Let Go(x) = \leax — me^xj
where a , /3 , l and m are as in (2.10) and p = 1, q — —1.
Define successively Gi, G2, ... by
G ^x) = G'q(x) = [laeax - m ^ x] ,
G2(x) = Gq(x) = [lc?eax - m ^ x\ ,
In general,
G„(x) = Gj,”>(x) = [/a"e“ - m /SV*],
It can be easily seen that when p = 1 , q = — 1,
Go(0) = a = Wo, Gj(0) = b = Wx and in general
G„(0) = [lan - m/3n] = Wn(a, b; 1, -1).
17
Horadam [9] extended the Fibonacci numbers to complex number field by
defining them as F* = Fn + iFn+\ . Berzsenyi [2] defined this by taking dif
ferent approach as a set of complex numbers at Gaussian integers and called
them as Gaussian Fibonacci numbers. He defined them as follows:
Let n and m be a non negative integer. The Gaussian Fibonacci numbers
F(n,m) are defined as F(n,m ) = J2 (T)ikFn- k where (n, m) = n + im are
the Gaussian integers and Fj are the (real) Fibonacci numbers. He proved that
F(n,m) = F(n — 1, m) + F(n — 2,m),n > 2. This relation implies that any
adjacent triplets on the horizontal line possesses a Fibonacci type recurrence
relation. In 1981, Harman [6] elaborated Berzsenyi’s idea and defined another
set of complex numbers by using the Fibonacci recurrence relation. He defined
them as follows:
Let (n, m) = n+im. where n,m E Z. The complex Fibonacci numbers denoted
by G(n,m ) are those which satisfy
(7(0,0) = 0, (7(0,1) = i, (7(1,0) = 1, (7(1,1) = 1 + i,
G{n + 2, m) = G(n + 1, m) + G(n, m), and
(7(n,m + 2) = G(n,m + 1) + G{n,m).
The initial values and the recurrence relations are sufficient to specify uniquely
the value of G(n, m) and for each (n, m) in the plane. It is easy to see that
G(n, 0) = Fn and (7(0, m) - iFm.
Harman’s definition has three fold advantages over Berzsenyi’s as given below:
2.5.7 Extension to Complex Fibonacci Numbers
18
1-In Berzsenyi’s definition any adjacent horizontal triplets in the plane satisfy
the Fibonacci recurrence relation where as in Harman’s definition any horizon
tal and vertical triplets are same.
2. Horadam’s Complex Fibonacci numbers F* come as a special case for Har
man’s. Indeed, F* = G(n, 1).
3. Harman was able to prove some new summation identities for {Fn}. Pethe
[18], in collaboration with Horadam extended Harman’s idea to define Gen
eralized Gaussian Fibonacci numbers. They again denoted these numbers by
G(n, m) and defined them at the Gaussian integers (n, m) as follows:
Let Pi, Pi be two fixed non zero real numbers. Define
(7(0, 0) = 0, (7(0,1) = i, G( 1,0) = 1, (7(1,1) = P2 + iPi with the conditions
G(n -f 2, m) = PiG(n + 1, m) — qiG(n, m), and
(7(n, m + 2) = P2G(n, m + 1) — g2(7(n, m).
Using this extension of Harman’s definition they were able to obtain vari
ous summation identities involving the combination of Fibonacci numbers and
polynomials, Pell numbers and polynomials and Chebyshev polynomials of the
second kind.For example it is proved that:2k
G(n + 2k + s,m + 2k + s) = bp( 1 + i) E { - iy q (2k~^Un+j+sUm+j+sj=i
2k+apq2 E (—l)JV 2fe-j)Un+j_2+sUm+j_i-t-s + q2kG(n + s,m + s)
j=i
where s = 0 or 1, and Un — VFn(0, l;p, q).
Putting different values for p, q, a and b, we get various identities involving the
Fibonacci numbers, the Pell numbers and polynomials etc.
19
Fibonacci numbers can be extended to Tribonacci numbers as follows:
Definition: Tribonacci numbers sequence Sn is defined by
Sn = PlSn-l + P2*Sn—2 + P3Sn-3i (2-14)
where n > 3 and Pi ,P2,P3 are arbitrary integers. Many scholars studied this se
quence by taking different initial conditions. Shanon and Horadam [22] studied
this sequence by taking following three sets of conditions.
2.5.8 Extension to Tribonacci Numbers
So — 0, Si — 1, S2 — Pi, (2.15)
So = 1, Si = 0, S2 = P21 (2.16)
S0 = 0, Si = 0, S2 = Pz- (2.17)
Denote the {S„} with condition (2.16) by {S*}.
One can observe that for Pi = 1,P2 = 1 and pz = 0, {S'*} reduces to {Fn} .
2.5.9 Extension of Fibonacci sequence using General
ized Circular Functions
The generalized circular functions are defined by Mikusinsky [12] as follows:
Let
oo fn r + j
J ^ (nr + j)\oo -fnr+j
NrJ® " 5 (nr + i ) !’
j = 0,1, •••, r — 1; r > l (2.18)
j — 0,1, —,r — 1; r > l . (2.19)
20
Note that
NitQ(t) = el, N2,o(t) = cosht, N2ti(t) = sinht and
Mifl(t)= e~l, M2fi(t)= cost, M2>1(t)= sint.
These functions are studied by Mikusinski and proved some of their basic
properties. Further studies of these functions are found in Pethe and Sharma
[17],
Results for generalized circular functions
Differentiating (2.18) and (2.19) term by term with respect to t, it can be
easily seen that
Mr,j-p(t), 0 < p < j,
Mr,r+j— p(t)> 0 — J ^ V — 'C-
(2.20)
JvffW =Nr,j-p(t), 0 < p < j,
Nr,r+j—p(t')i 0 — 3 <~ 3 <• P — T-
Particularly in (2.21) note that
JV$<t) = Nr,o(t).
(2 .21)
In general
Further note that
N $ )(t) = Nr<Q(t) ,n > l. (2.22)
Nr,o(0) = N $ \ 0) = 1
21
Observe that from (2.21) for a fixed value of r the functions consti
tute a fundamental system of differential equations.
y (r) + Y = 0 (2.23)
such that
M-j}(0) = 6pj , (p ,j = 0,1,..., r - 1). (2.24)
S.P. Pethe and C.N. Phadte [19], studied generalized Fibonacci functions using
the property of circular functions. They defined it as follows:
Definition: Let
Po(x) = ^ [ lN Tfi(a*x) - mNrfi(l3*x)] (2.25)
where r is a positive integer and
a* = a 1/r,p* = P1/r, (2.26)
a, 13 being distinct roots of x2 — px + q = 0 and Z, m, and d are as defined in
(2.10). Note that a + 13 = p, a{3 = q.
A sequence of generalized Fibonacci function {Pn(a;)} is defined as follows:
Pi(x) = Por)(x),
P2(x) = Pq t){x),
and in general,
Pn(x) = P t \ ^ , n > l .
Then from (2.25) we get
Pi(x) = ^ [ la N rto{a*x) - m/3Nrfi{j3*x)}
22
A(x) = ^ [ la 2Nrfi(a*x) - m/32Nrfi(/3*x)}
In general, Pn(x) = -^[lanNrfi(a*x) - mPnNrt0(/]*x)].
Observe that for r = 1, a* = a, /3* = /3 and Nrfi(t) = Nlfi(t) = el with
p = 1, q = — 1 and r = 1, becomes
P"(X) = 2^ a "e"* “ mfine^x] = Gn(x). We have the following.
Theorem 2. Pn(x) satisfies the recurrence relation
Pn{x) = pPn_j(x) - qPn_2{x). (2.27)
For simplicity , let QntT(l,a ‘,x) = lanNrfl(a*x) with the corresponding ex
pression for Qn,r(m,/3; x).
Note that Pn(x) = '^j\Qn,r(l>&]X') Qn,r(pi> fi] x')}.
The various identities of Pn(x) are listed below:
i) E Pk(x) = Pn+2^ ~ P l^ ~( p ~ 1)[p"+i(x) ~ Afo)]
f / , xfc P /_% (P + l)[po(i) + (—l)n+2Pn+1 (x)] - Pi (a:) + ( - l ) n+1Pn+2(x) n j - M i j n W - (p + g + 1)
iii).Pn_i(x)Pn+1(x) - P2(x) = Q n ,r (J i X Qn r TTl, /9, x)
iv) -P„ {x)E*+1 (x)-qP n- i (x)F£ (x) eaXQn+sAh x ) - e/3XQ n + s ,r (m , fi\ x) 2d
\ t~i / \ n* / \ r> /■ \ rr>*/ \ &aVQn+s,r(l> U) A* QnjrS,r{m‘i U)v).Pn(u)E*s+1(v) - gP„_i(n)P;(n) = --------------------- —---------------------
i).P„2(x) - gPn2_: (x) = --------^ --------------------
23
vii).p2+1(x) _ q2Pl_l{x ) = ER h S l' a;x>> Ql,r(m,P;x)]4 q
Viu).Pn+1- s(x)Pn+1+s(x)-PZ+1{x) = lQ n+ lA ^x)Q n+ lA ™ ,M ]l2 <* ' F <**0 °]4 a?
ix)-P„(x)P„+l+t(x)-P„-,(x)P„+i+nl(x) = {p«A ‘,*;x)P*Arn,l}-,x)(a- P‘){a‘+M ft4 (Pqs
All these properties can be proved by using Binet formula.
2.6 F ibonacci Num bers and B inom ial Coeffi
cients
There is an interesting connection between Fibonacci numbers and Binomial
coefficients [26]. We get the following pattern if Binomial coefficients are ar
ranged in a triangular arrays as follows:
/
V
/ \ / \1 1
v0 M1)M M M\° A 1 A 2;
/ \ ( _ \
V1 / v2;
\
/
24
( . \
V 0 /
( . \ ( . \
V 1 A 2 /
/ \
v 3 /
( . \
v 4 /Table B: Binomial coefficient in triangular array
Above pattern is also called Pascal Triangle which is listed below:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Table C: Pascal Triangle
We left align the above triangular arrangement can be listed as below.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Table D: Triangular array of numbers
25
Lines connecting the number of triangular array shown in Table D are
called ascending diagonals. We observe that the sum of the numbers on
the ascending diagonals are Fibonacci numbers. We prove this fact as
follows:
It can be observed that each of the first two ascending diagonals consists of
number 1. i.e., Fq and F\. In order to prove the result, because of the relation
(2.1), it is sufficient to show that the sum of all numbers making up (n — 2)th
and (n — l) th diagonal of the triangular array is equal to the sum of numbers
lying on the nth diagonal.
The (n — 2)th diagonal consists of the numbers
\n-3 ' (n-3)/2 ^
0 , V 1 ,
*•*?v (n-3)/2 ,
if n is odd and
\ ( \ ( , , , \n-3 n-4 (n-4)/2
0 ,
1
< 1 > (n-4)/2 J
if n is even.
The (n — l) tfl diagonal consists of the numbers
\n-2 I n F
/ \ (n-2)/2
0 >?
, 1 y
’"J
to "to
if n is even.
The (n — \ ) th diagonal consists of the numbers
26
/ N / \n-2 n-3
. 0 y
>
, 1 y
' (n-l)/2 '
, (n-3)/2 tif n is odd. Hence,
the sum of numbers of the (n - l) th and (n - 2)tfc diagonals is
/ \ (n-l n-2+
\ 0 , < 1
+ ... +(
\
(n+l)/2
(n-3)/2
\ / \(n-l)/2
+(n-l)/2
if n is odd and
M *
n-2 ^+ ...+
/ \(n+2)/2
+
/ \ (n)/2
° J V 1 , (n"4)/2 y (n-2)/2 j
if n is even. We use the fact that
V
w
= 1
and
( \ ( \ (k k k+ 1
+ —
< < i i + 1 J i+1
The sums that are obtained are the numbers that lie on the nth diagonal of the
triangular array if n is odd or even. This proves the theorem. British mathe
matician Ron Knott [31] provided interesting insights into Fibonacci numbers.
He found Fibonacci numbers as the sum of “rows” in the Pascal triangle. The
arrangement of numbers by drawing Pascal triangle with all the rows moved
27
over by one place shows the Fibonacci numbers as the sums of columns shown
in the table below:
Table E: Fibonacci numbers as sums of columns
28
Pascal’s triangle is drawn with all rows moved over by one place. It clearly
shows the Fibonacci numbers as sums of columns as shown in the last row.
2.7 D ivisib ility properties of Fibonacci Num
bers
Here we list some results on divisibility of Fibonacci numbers. Divisibility of
Fibonacci numbers is important when we try to study the periodicity of the
sequence.
Theorem 3. I f n is divisible by m then Fn is divisible by Fm.
Proof. Let n be divisible by m. Then n = mmi, where mi is a positive
integer < n. We prove the theorem by induction on mj.
If mi = 1 then n = m and the result is obivious. Assume that the result holds
for mi = k. Therefore, Fmk is divisible by Fm. Now consider Fm(fc+i). Using
property (1) of section (2.3), we get
Fm(k+1) = FmkFm+1 + Fjnh—iFfn,.
The right hand side of this equation is divisible by Fm. This proves the theo
rem. ^
Theorem 4. : Consecutive Fibonacci numbers are relatively prime,
i.e., [F„,F„+i] = 1 .
29
Proof. If Fn and Fn+\ have same common divisor d > 1 then
Fn+1—Fn = Fn_! is divisible by d. We can prove by induction that F„_2, Fn_3,...
and finally Fo is divisible by d > 1. This contradiction proves the theorem. □
The following properties are easy to prove.
(a) A Fibonacci number F„ is even if and only if n is divisible by 3.
(b) A Fibonacci number Fn is divisible by 3 if and only if n is divisible by 4.
(c) A Fibonacci number Fn is divisible by 4 if and only if n is divisible by 6.
(d) A Fibonacci number Fn is divisible by 5 if and only if n is divisible by 5.
(e) A Fibonacci number Fn is divisible by 7 if and only if n is divisible by 8.
(f) There is no Fibonacci number that gives the remainder 4 on division by 8
and also there is no even Fibonacci number divisible by 17.
30
Chapter 3
Pseudo Fibonacci Sequence
3.1 Introduction
It is well known that Fibonacci sequence has been generalized in many ways to
generate a new sequence. These generalizations are based on either changing
the coefficients in the homogeneous recurrence relation defining the Fibonacci
sequence or by changing the initial or seed values. In this chapter we define a
new sequence using a non homogeneous recurrence relation which gives rise to
a generalized Fibonacci Sequence. We call this sequence as pseudo Fibonacci
sequence.
Definition: The Pseudo Fibonacci Sequence {gn} is defined as the sequence
satisfying the following non-homogeneous recurrence relation.
fl'n+2 = 9n+1 + 9n + Atn, n > 0 and t 7 0, CHI, Pi (3-1)
with go = 0 and g\ = 1. Here t is a fixed parameter for the relation (3.1) .
The first few terms of {#„} are:
g2 = l + A, g3 = 2 + A + At, g4 = 3 + 2A + At + At2 and so on.
Observe that each pseudo Fibonacci number is such that its first term is a
Fibonacci number and remaining terms form a polynomial in t with coefficient
31
A times a Fibonacci number, pseudo Fibonacci Numbers for negative indices
are given by the non-homogeneous relation <?_„ = 9 - n+2 — 9-n+1 — At~n.
As in case of all the extensions of Fibonacci sequence, we can obtain Binet
type formula for pseudo Fibonacci sequence in the usual way.
Binet type formula for {gn} is given by
9n = ci a? + c2/?" + ptn, (3.2)
where ai and Pi are the roots of characteristic equation x 2 — x — 1 = 0.
The constants Ci, c2 and p are
_ [(1 - p { t - P i ) } _ \p(t - a i) ~ 1] A1 a i - Pi 2 a i - Pi ’ (t2 - t - 1 ) '
Binet type formula is useful in developing the theory of pseudo Fibonacci
sequence. We shall use it in the subsequent sections of this chapter to obtain
various identities.
3.2 Som e Fundam ental Identities o f {gn}
Many interesting identities may be derived for the sequence {gn]- Some of the
identities are given below.
i) The G enerating Function of {^n} is given by
Z(x) = , ^ + 1~ 2V provided |t| < 1.(1 - t)(l - x - x2)
32
Proof. Let Z{x) = £ gnxn. Then,fc=0
z {x) = go + g\x + g2x2 + ... + gkxk + ...
xZ{x) = g0x + gi x2 + g2x3 + ... + gkxk+1 + ...
(1 + x)Z(x) = g0x + (gx + g0)x + (g2 + gi)x2 + ... + (gk + gk_lXk + ...
= 9o + (92 ~ A)x + (g3 - at)x2 + ... + (gk+1 - Atfe_1)xk 4- ...OO
x ( l + x ) Z ( x ) = go + g2x2 + g3x3 + ... + gk+1xk+1 + ... - A Y ^ ik0
OO
= z(x) - g i x - A j 2 t k-
Since go — 0 and pi = 1 , we get, Z(x){x2 + x — 1) = —g\x — £ tk.k= 0
Hence
Z(x) =x + 1
1 - 1(1 — X — X2) ’
x(l — t) + 1(1 — t){ 1 — X — X2) , provided |t| < 1.
ii) E gk = gn+2 - gl - A E t lk=0 k=0
Proof The recurrence relation (3.1) can be written as
□
9k = 9k+2 — 9k+1 — Atk.
33
Adding up the equations for k = 0, 1 , ...,n term by term, we obtain
n n+2 n+1 n
fc=0 fc=2 fc=l fc=0n n n
= ( E 9 k -g o ~ ff i+ gn+1 + gn+2) - ( E 9k - go + gn+\) - A E*=0 fe=0 k=0
n= 9n+2 - fl-l ~ A 5 3 <*•
k=Q
Hence the result. □
hi) E g2k-l = g2n ~ g-2 - A £ t 2k~2. k=0 k=0
Proof. From equation (3.1) we write
92k—1 = 92k ~ 92k~2 ~ At2k~2.
Adding up the equations for k = 0,1,..., n term by term, we obtain
5 3 92k—1 = 5 3 52k ~ 5 3 £,2k-2 - a 5 3 i2fe-2,fc=0 k=0 k=0 fc=0
= 5 > * ~ n£ g 2 k - A ± t 2k- 2, fc= 0 fc= —1 fc= 0
= l?2n - 9 - 2 - A E ^ fe~2- fe=0
□
iv) E g2k = g2n+l - g-1 - A E t 2k_1.k=0 k=0
Proof. Recurrence relation (3.1.1) can be written as
a j.2n—l92n = 92n+1 ~ 92n-1 ~ At
34
R epeated u se o f th e above equation for k = 0 , 1 , 2 , n and sum m ing up leads
to,
X^ 92k = 5 3 92k+i — ^2 52fc-i — A ^2 i 2fe~ \fc=0 fc=0 k—o k=0
— XT 92k+i — X I 52*1+1 ~ A X ] i 2fe_1,k=0 k——1 *=0
= 92n+\ ~ 9\ ~ A j 2 t 2k~l ■*:=0
□
V) E g3k -2 = 2 {gSn - g —3 + A (1 + t ) E t 3k~3}. k=0 0
Proof. From th e recurrence relation (3.1) we write,
29n — 9n+2 — 9n- 1 + A t” X( l + t) . (3-4)
For n = - 2 , 25_2 = 50 - 5-3 + A t~ 3( l + 1),
For n = 1, 2gx = gz - g0 + A t° ( l 4 - 1),
For n = 4, 2g4 = 5e ~ 53 + A t3( l + 1), . . .
For n = k, 2gk = 5fc+2 - gk- i + A tfc-1( l + i) .
Adding th e le ft hand side term s and right hand side term s separately, we get
2 E 53fc—2 = E 53k ~ E 53*: + A (1 + t) E ^ ' 3-k=0 fc=0 * = -l 0
Hence,
E 93k—2 = \{93n — 5 -3 + A(1 + 1) E t 3/5~3}- k=o o
□
Vi) E g3k-l = |{g3n+l - g—2 + A(1 + t) E t3k_2}- k=0 0
35
Proof. R ep eated use of equation (3.4) for k = 0 , 1 , 2 , . . . , n gives the following
equations.
For n = -1, 2g_! = ^ - g_2 4- A r 2( l + f),
For n = 2, 2g2 = - 5i + A*x( l + i) ,
For n = 5, 2g5 = g7 - g 4 + At*(l 4 -1), . . .
For n = k, 2gk = gk+2 ~ 9k- i + A**-1( 1 4-1).
Adding th e left hand side term s and right hand side term s separately, we get
2 E 93k-1 = E 53*+i — 2 93k—2 + A(1 + i) E*3fe-2-k=0 fc=0 fc=0 0
Hence,
E 93k-\ — 2{93n+l — 9-2 + A (l 4- t) E ^ 3*-2 }- □fc=0 o
Vii). E g3k = §{g3n+2 ~ g-1 + A(1 + t) E t3k_1}. k=0 0
Proof. R ep eated use o f equation (3 .1) for k = 0 , 1 , 2 , . . . , n gives the following
equations.
For n = 0, 2g0 = 92- 9-1 + A*_1( l 4 - 1),
For n = 3, 2g3 = 95 ~ 92 + At2( 1 4 -1),
For n = 6, 2g& — g&-g^ + At5( 1 4 -t), . . .
For n = k, 2gk = gk+2 - gk- 1 4- A tfc_1( l 4-1).
Adding th e left hand side term s and right hand side term s separately, we get
2 E 93k = ~ 9 - l + 93n+2 + A (1 4 -1) E t3k~l . k=0 0
Hence,
E 93k = \{g3n+2 - g-i + A( 1 4- 1) E*3*'1}. k=0 0
□
36
viii) £ © g n -k = g2n - p[t2n - (1 + t)“], where p = -- ^ k=0 ' — t — 1
Proof. Using Binet type formula (3.2), we get
= ci(l + m)" + c2( 1 + &)" + p{ 1 + t)n
= cia?" + c2 n + pt2n - pt2n + p( 1 + f)n
= 92n-p\t*n - ( l + t ) n}.
ix) E (k)g3k = 2n(g2n - p t 2n) + p ( l + t 3)n, k=0 v /
Proo/. Here we use the relation af + 1 = 2a2 which is obtained from charac
teristic equation a2 = au + 1 . We have
= c i( i+ ai r + 02(1+ p i t + p( i + t3r
= c1(2a2)n + c2(2/32r + p ( l + t3r
= 2 n (Cla 2n + c2P ln + p t2n) - 2np t2n + p( 1 + t 3)n
= 2ng2 n - 2 np t 2n+ p ( l + t3)n
= 9 2 n - p [ t 2 n - ( l + t n
l n —fe
Hence the result. □
where p = At2 - t - 1'
37
□
Sim ilarly, w e have the following result.
x). E (k)g4k = 3n(g2n - pt2n) + p (l + t4)n],k=0 'where p = —-------------.v t 2 - t - 1
Proof. T h e p roof is similar to th e above except th a t w e u se th e relation, af +
1 = 3 a \.
t ( * ) s « = t (* ) (C l< + ^ + pt«)
= Cl( i + a i r + C2( i + f i r + p (i + t4r
= ci (3a2)" + c2(3 01Y + p(l + t4T
= 3"(Cla 2n + c2Pin + pt2n) - 3npt2n + p{ 1 + t4r
= T g 2n - 3 npt2n+ p(l + t4)n.
Hence th e result. □
w here T n = E t 2k+1k=0
Proof. W e prove th is result by m eth od o f induction.
R esult h o ld s true for n = 1.
A ssum e th a t it is true for all values o f k from 1 to n i.e .,n—1
„ M " +1 + 9n+ltn - 1 - A E t2k+1]E 9ktk~l = --------- fc"°fc=0 (t2 + t - 1)
38
Now to prove th a t it is true for n = k + 1
i.e., to prove
E 9ktk~lk=0
[9n+itn+2 + gn+2tn+1 - 1 - A £ t2k+1] _______________________ fe=0____
(t2 + 1 - 1)Consider
n+l n£ 9ktk~x = £ 9ktfc_1 + 9n+ltnfc=0 k=0
n—1[9ntn+1 + 9ntn ~ 1 ~ A E t2k+1]
k=0(i2 + 1 - 1) + 9 n + l t n
n—1[</n*n+1 + s„+1r - 1 - i E t2k+1] + gn+1tn( f + t - 1)fc=0
(t2 + i — 1)n—1
[ ( S n * "+1 + 5 n + l< n + 1 ) + 5 n + l< n+2 - l - A Y , t 2k+ l]k=0
(i2 + t - l )n — 1
[Sn+2*n+1 - A t2n+1 + 5n+l<n+2 - 1 - A E t2k+1]_______________________________ fc=0____
i f + t - 1)
Hence,
E 9kt‘k=0
fc-1 _ \gn+1tn+2 + gn+2tn+1 - 1 - ATn\(t2 + t - 1)
T his com pletes th e proof. □
Xii) E gk = gngn+l - A E gktk k=0 k=0
Proof. L et u s prove it by induction over n. For n = 1, identity (xii) takes the
form So + 9i = 9i92 ~ A(gQt~x + Si) , which is true.
A ssum e th a t it is true for n — k i.e .,
£Sfc = SnSn+i ~ A j 2 9 k t k X-fc=0 fc=0
(3.5)
39
We now prove that (3.5) is true for n — k + 1.
Adding g^+1 to both sides of the above equation gives
n+l nY 9k = gn9n+1 - A J 2 + 9n+lk=0 k=0
n= 9n+l[9n + 9n+l] — A ^ S f c ^ ” 1
fe=0
= 9n+l[9n+2 ~ A tn] ~ A J 2 9ktk~Xk=0
n+1~ 9n+l9n+2 ~ A ^ gktk~X,
k=0
which proves that identity (xii) holds true for n = k + 1 .
This completes the proof. □
3.3 A G eneralization to a new Sequence {Gn}
In this section, the sequence {gn} is extended to a generalized form defining a
new sequence {Gn}. It is defined as follows:
Definition: A generalized pseudo Fibonacci Sequence {Gn} is the sequence
satisfying the following non-homogeneous recurrence relation.
Gn+2 = pGn+i — qGn + Atn, n > 0, A ± 0 and (3.6)
with
Go = a and Gi = b. (3.7)
Here a, b, p, q are arbitrary integers and a, /3 are the roots of x2 — px + q = 0 .
First few generalised pseudo Fibonacci numbers are given below:
Go = a, G\ = b,
40
G2 = (pb — qa) + A,
G3 = (p2b — pqa — qb) + Ap + A t,
G4 = (p36 - p2qa - 2pg6 + #2a) + A(p2 - ^) + ylip + At2, and
G5 = (p46 - p3ga - 3p2qb+ 2q2a + q2b) +A(p*~ 2pq) + At(p2 - q)p+ At2p + At3.
Observe that each generalized pseudo Fibonacci number Gn, n> 2 consists of
two parts. The first part contains expression in p, q, a ,b and the second part
is a polynomial in t whose coefficients are A times the terms in p and q. This
is shown in the following tables.
n Expression
2 pb — qa
3 p2b — pqa — qb
4 p3b — 2pqb — p2qa + q2a
5 p4b — 3p2qb + q2b — p3qa + 2 q2a
6 p5b — 4p3qb — p4qa + 3 q2a + 2 q2
Table F : First part of G„
41
n A to At3 At4 At5 At6
2 1
3 P 1
4 p2 - q P 1
5 p3 — 2pq p2 - q p 1
6 p4 - 3p2q + q2 p3 — 2pq p2 — q P 1
Table G: Second part of Gn, n ^ 2
From the above tables it can be concluded that Gm, the mth term of gen
eralized pseudo Fibonacci sequence is given by
Gm = Wm(a,b;p,q) + A £ Wfe(0,k=1
where Wm is mth term of the extended Fibonacci sequence or Horadams se
quence listed in Chapter 1. section (2.5.1). Thus we have
TO— 1Theorem 5. Form > 2, the term Gm — Wm(a, b-,p, q)+A £ VFfe(0, l;p,
k=1
of the sequence {Gm} satisfy the non-homogeneous recurrence relation
Gm+2 = pGm+1 — qGm + Atm.
Proof. Consider
’E 1 Wk{ 0,1 ;p, q)tm~k+1 = Wxtm + W2tm~x + W3tm~2 + ... + Wm+ik=1
=Wxtm + (pWl - qWo)tm~l + (pW2 - qWi)tm~2 4-... + (pWm - qWm- 1)
=Witm + p(W1tm~1 + W2tm~2 + ... + Wm) - q{Wotm- 1 + Witm~2 + ... + Wm- 1)TO . TO—1 ,
=Wi*m + p £ Wfe(0, - g £ W*(0,fc=l fc=o
42
That is,
m+l£*=i£ Wk(0,l;p,q)tm- k+1) =
m m—1+ p £ w*(0,1; p, g)tm- fc - g 53 h m o , i ; p, g ) r - fc- 1 (3.8)
fc=l fe=0
Now
fc=i
m+lGm+2 = Wm+2(a, 6;p, q) + A £ wk(0,1;p, q)tm~k+1.
k=lUsing equations (3.6) and (3.8),
Gm+2 = pWm+i(o, b;p, q) - qWm(a, b;p, 9) + Ap 53 Wk{0, 1; p, q)tm~kk=1
m—1- g 4 £ Wlfe(0, l ; p ,g ) r - fc- 1 + A W
= p[Wm+i(a, b;p,q) + A £ Wfc*m"fc]fc=i
TO—1- g[W(a, 6;p, g) + A 53 Wfc(0, l;p , g)*”1-* '1] + At"
k=0
—pGm+1 — gGm + -df”
Hence the theorem. □
3.4 Som e Fundam ental Identities o f Gn
In this section we obtain some fundamental identities {Gn} which are useful
in further development of the subject.
(i) B inet T ype Formula: Let the homogeneous relation corresponding to
the equation (3.6) be given by
Hn+2 = pHn+1 - qHn (3.9)
43
If a and j3 be the roots of x2 — px + q = 0, the characteristic equation of
(3.9), then
with the initial conditions as of Gn, i.e. Hq = a and H\ — b.
a —P + d „ _ p - d2 ’ P 2 (3.10)
where d = y/p2 — 4q ^ 0.
Note that
a + = p, a(3 = q, a — (3 = d. (3-H)
We now obtain the Binet type formula of generalized pseudo Fibonacci
sequence.
Theorem 6 . For every n E N, the Binet type formula of G PF
Sequence is given by
Gn = Cia" + c2/3n + z tn, (3.12)
where
(b — a/?) — z(t — P) (aa — b) — z(a — t)Ci — ------------------------ , c2 = —a —13 a — j3A
t 2 - p t + q (3.13)
Proof. Let G ^ = ztn be the particular solution of (3.6), hence
ztn+2 = zptn+1 — zqtn + Atn, which gives 2 = —---------- .t2 —pt + qHence particular solution is
Atnt2 —pt + q
(3.14)
44
C1 + C2 = z — a and Cia + c2fi — b — zt.
Solving these equations for c\ and c2 , we get,
Using initial conditions we get,
(b — aft) — z{t — P) _ (aa — b) — z(a — t) a — [3 ’ °2 a — (3
Hence solution of (3.9) is
(3.15)
HW = c1cr + c20n (3.16)
and the general solution of (3.6) is given by
Gn = HW + GW = Ci a" + c2pn + ztn
which is as required. □
Note that
C1+C2 = a—z, ci— C2 = ((2b—ap)—z(2t—p))d~1, C1C2 = edT2 (3.17)
where e = abp — b2 — a2q — z{bp — 2bt — 2aq + atp + A}. Gn in terms of
Un can be given as
Gn - bU„ - aqUn- 2 + (ztn - ztUn- 1 + zqUn- 2)
Qn+1 _ ftn+1where Un = --------- -— is the generalized Fibonacci number [7] whicha — psatisfies the recurrence relation
Un+2 — pUn+i qUn.
45
(ii) Generating function:
Generating function G*(x) for G„ is given as
G*(x) 1(1 — px + qx2)
Ax2 1 — tx + (a + bx — apx) provided \tx\ < 1 .
(3.18)
Proof. Let00
G*{x) = Y ,G n xn. (3.19)n =0
We obtain px~1G*(x) , x~2G*{x) and subtract them from G*(x). With
the use of recurrence relation and initial conditions, we get rid of all
summands except the first two. This gives the following:
x~2G*(x) - px-'GTix) + qG*(x) = £ (£ *+ 2 - pGn+\ + qGn)xnn=0
+ G qx~2 + G \x ~ l - pG ox- 1 ,
G*{x){x 2(x) — px 1 + q] = 2 Atnxn + ax 2 + bx 1 — pax 1.n = 0
Therefore,OO
A E t nxn+2 (a + bx_ am)G*(x) = „ n=0 ------ ^ + 1 +
(1 — px + qx2) ' (1 — px + qx2)Hence,
^ ^ (1 — px + qx2)Ax2
1 — tx+ (a+ bx — apx) , provided |tx| < 1 .
□
Generalized pseudo Fibonacci sequence Gn , like Fibonacci sequence, has
following properties.
(iii) lim = a , if \t/a\ < 1 .v ' n—>oo u n -1
46
(iv) lim = ak , if \t/a\ < 1 .V ' n ->00 n - k 7 1 ' *
We now look at some of the important identities involving sums of the terms.
Proposition 7. (a) The sum of first n generalized pseudo Fibonacci
term s is given byn—1 1E Gk = ?------------
k=o (p - q - 1)n—1
Gn+i - b - (p - 1)(G„ - a) - A £ t kk= 0
(bJThe sum of first n generalized pseudo Fibonacci Sequence term sn-l
with a lternating signs is given by £ (—l)*Gk =k=0
1 [(—l ) n+1G„+1 - b + (p + 1 ){(—l)n+1G„ + a} + A nE ( - l ) ktk(p + q + l j L k=o
Proof, (a) Using Binet type formula (3.12), we write
qGk — pGk+1 — Gk+2 + Atk.
Adding the left hand terms and right hand terms separately, we obtain
n—1 n—1 n—1 n—1Q E Gfc = P E Gfc+1 — E Gfc+2 + A X) fc>
fc=0 fc=0 fc=0 fc=0n n+1 n—1
= p E g * - E g ‘ + ^ E * ‘ -k=1 k=2 fc=0
= P T .(G k — Go — l-Gn) — X (Gfc — Go — Gi + Gn + Gn+1) + A X) tk.k=1 fc=o fc=o
Therefore,
n—1
Ek=0
n—1
fc= 0E Gfc(9 — p + 1) = p{Gn — Go) + Go + G\ — (Gn + Gn+i ) + A X! t
— ip — 1)(G„ — Go) + (GiGn+i) + A XI tk.
Hence,
E G* =fc= 0
1
i p - q - 1)Gn+1 — b — (p — 1 )Gn - a
fc=0 J
47
. (b) Writing the equation (3.6) as
qG/c- i — —Gfe+i + pGk + Atk 1
and substituting the values of A; = 1, 2, n leads to
qGq — —C?2 + pG\ + At0,
—qGi — (?3 — pG2 — A t1,
9G2 = —G4 + pG$ + At2,
In general, for nth term
= (—l) nGn+1 + ( - 1 )npGn +
Taking the sum of the terms separately on both sides , we get
Q 1 )kGk — (p + 1)(—G2 + G3... + (—1)" 1Gn + (—l ) nGn+\ + pG\k= 0
n—1+ A j 2 ( - l ) ktk-
k= 0
Hence ,
ip + q + 1 ) E ( - l )kGk = ( - 1 )n+1Gn+1 - G l + (p+ l ) { ( - l ) " +1Gn + Go}fe=0
+ A ]T ( - l) fct fe.
Therefore,
e V i ) ^ =1
(p + q + l)( - l ) n+1G n+i - b + (p + 1){(—1)"+1G„ + a} + / e ( -1 )ktk
k= 0
□
Next result is about the sum of the product of Gk with powers of t.
48
Proposition 8. £ Gktk_1k=0
= ( i ^ p t + V ) [a/t -ap+b - - »tG»)+AS t2k+i. -Proof. Using recurrence relation (3.6), we write
Gnf1- 1 = [pG„_! - qGn- 2 + Atn- 2] r - \
Gn-!tn- 2 = [pG„_2 - 9Gn_3 + Atn~3]tn~2,
G2tx = \pGx - qG0 + At°)tK
Adding terms on both sides, we get
£ GkP-1 = p J : Gktk - q "e Gktk+l + A ”£ t2k+1.k= 2 * = 1 k= 0 fe=0
Therefore,
£ Gfct* - 1 - Got" 1 - Gx = p £ Gfctfc - p(G0 + Gntn)k=0 fc=0
- q £ Gfcifc+1 + 9(Gn_!tn + Gn£n+1) + A 2 t2fe+1A)=0 fc=0
Go£ Gfctfc X(1 — pt + qt2) — + Gi — pGo — (pGn — g(G„_i)t"t_rv t*=0
+ qGntn+1) + A J 2 t2k+\k=0
Hence,
£ Gktk=0
fc-1 _ 1n (j - a p ) + b - t n(Gn+1-q tG n) + A n£ 1t2k+1(1 —pt + qtJ) t k=o .
We now obtain the sum of the squares of Gn and sum of the product of
two consecutive G„.
49
For simplicity, let X = E G|, Y = E GfcGfc+i, Si = E Gfet fe_1 andfc=0 &=0 fc=0
S2 = E t2k- Let t>! = 1 — p2 — g2 and v2 = 1 + p2 — q2. Also letfc=0
Ti = (1 - P2)[G2 - G2+1] + (G2 - G2n+2) + 2A(G0t~2 + Gxt~'
- Gn+1tn~l - Gn+2tn) and
T2 = (1 +P2)[G2 - G2n+1] + (G2 - G2n+2) - 2p(Gn+1Gn+2 - G0G1).
We have the following:
Proposition 9. (a) The sum of squares up to n+1 generalized pseudo
Fibonacci terms is given by" 2 T , + T2 + 2AS1( f 1 - q2t) + A2S2(q - 1) + (T3 - T4)q
k=o k Vx + qVa(b) The sum of products of two consecutive generalized pseudo Fi
bonacci sequence numbers up to n + 1 terms is given byA „ ^ (T, + T2)V2 - (T3 - T4)V, + 2 A S ,( t 1V2 + tqV l) - ASS2(V, + V2)E G kGk+1--------------------------------------- 2p(VI + qV2)
Proof. From the recurrence relation (3.6), we write
G l+ 2 - p2G 2n+1 = (G n + 2 - pGn+l)(G n + 2 + pGn+l)
= (Atn - qGn){pGn+1 - qGn + Atn + pgn+i)
= (Atn - qGn)(2pGn+1 - qGn + Atn)
= 2pGn+1Atn - 2AqGntn + A2t2n - 2pqGnGn+1 + q2G2n
= 2{Gn+2 - A tn)Atn + A2f2n - 2pqGnGn+1 + q2G2n
= 2AGn+2tn - A2t2n - 2pqGnGn+i + q2G2n.
50
Adding the left hand side terms and right hand side terms separately, we get
E Gl+2 ~ P2 E Gl+1 =fc=0 fc=o
2A E Gk+2tk — A2 t2fc — 2pq E GfcGfc+i + g2 E G\fc=0 k=0 k—Q k=0
^ 2 Gl — G0 ~ Gl+ Gn+1 + Gn+2 - P 2 E GI - P2(Gn) =fc=0 fc=0
2A E Gfc+2t* - A2 E t2k - 2pq E G*Gfc+1 + g2 E G 2 - p 2(G2+1 - G2)fe=0 fc=0 fc=0 k=0
w+2= 2A E Gktk~2 - A2 E t2fc - 2W E GfcG*+1 + g2 E G2.
fc= 2 fc= 0 fc= 0 fe= 0
Hence,
(i - ?2 - p 2) £ o i = g? - a \ - &M - g«+2 +p»(G5 +1 - eg).fe=0
+2A* -1 E G*tfc_1 - 2A(G0r 2 + Git" 1 + Gn+xf1- 1 + Gn+2tn) - A2 E t2fe -fc= 0 k= 0
2pq E GfcGfc+ik=0
Therefore,
(1 - «2 - P2) S GJ = (1 - p’XCS - Gin) + (G? - G?+2) + 24t-» E Gktk-1k=0 fc=0
+2A(G0r 2 + G it-1) - Gn+1r - x - Gn+2tn) - A2 E t2fc - 2pq £ GfcGfc+i.fc=0 fe=0
Above equation can be written as
VxX = 7\ + 2A t-1 Si + T2 - 2pgY - A2S2. (3.20)
r- «>+51
Similarly,
^n+2 — Q2(*n — {Gn+2 — QGn)(Gn+2 + qGn)
— {Gn + 2 + ( ( ? n + 2 — p G n + i — Atn)(pGn- i + A tn)
= (2Gn+2- p G n+l - A tn)(pGn- i + A tn )
= 2pGn+2Gn+1 + 2AGn+2r - P2G2n+1 - APGn+1tn - ApGn+1tn -
= 2pGn+\Gn+2 + 2AGn+2tn - p2G2n+1 - 2ApGn+1tn - A2t2n
= 2pGn+1Gn+2 + 2A(Atn - qGn)tn - p2G2n+1 - A2t2n
- 2pGn+\Gn+2 + 2A2 - 2AqG„tn - p2G2n+1 - A2t2n
= 2pGn+lGn+2 - 2AqGntn - p2G2n+1 + A2t2n.
Hence,
G2+2 - q2G2n - 2PGn+1Gn+2 - 2AqGntn - p2G2n+1 + A2t2n.
Adding the left hand side terms and right hand side terms separately, we obtain
t g i+2 - e t g ifc= 0 fc= 0
= 2p t Gk+1Gk+2 - 2Aq E Gktk - P 2 E G\+l + A2 E t2kk—0 fc=0 fc=0 fc=0
i.e.,n +2 „ n£ Gl - q2 E G?
fc= 2 fc= 0
Tl+1 fl 71+1 712p E GfcGfc+1 - 2Aqt E G ^*"1 - P2 E G2 + A2 E i2fc-k=l fe=0 fc=l fc=0
Hence,
£ Gl - Gl - G? + GJ+i + GS+2 - 92 £ =fc= 0 fe= 0
2p ^2 GkGk+1 — 2pG0Gi + 2pGn+xGn+2 — 2Aqt ]E fc-ik= 0 fc=0
- p2 e ^ + p2^ - p2g^ i + x : ^ -jfc=0 k=0
A2t2n
52
(1 ~ q 2 + P 2) Y G 2k =fc=o
Therefore,
Gl + G\ G\+1 — G„+2 + GkGk+i — 2p(GoGi — Gn+.Gn+2)fc=0
2 A q tJ ^G ktk-k=0
1 + P2{Gq — G„+1) + 2A2 ^2 t2k.k=o
Hence,
(1 - q2+p 2) Y G 2k =k=0
(1 + P2)(Gq — G%+1) + (Gl — G^+2) + 2p 2 GkGk+ik—0
2p(G0G1 - Gn+1Gn+2) - 2Agt £ Gktk~l + p2(G20 - G2n+1) + 2A2 £ t2kfc=0 k=0
which can be written as
y2X = T3 + 2pY - 2AqtS1 - T 4 + a 2s 2. (3.21)
Solving equations (3.20) and (3.21) for X and Y, we get n T ,+ T 2 + 2AS1(t~1 - qH) + A2S2(q - 1) + (T3 - T4)q
k=o k Vi + qV2
and
Y GkGk+i —k=0
(Ti + T2)V2 - (T3 - T4)Vi + 2A S.it-1 V2 + tqV.) - A2S2(V. + V2)2p(V. + qV2)
□
Next we obtain sum of the even and odd indexed terms. Let E — Y G2ii=l
and O = Y G2i- i . Denote EA = A Y t2t and 0 A = A Y £2l_1 so thati=1 i=1 i=1
2nEA + Eo = A Y & ■i=1
53
Proposition 10. (a) The sum of the even indexed terms of {G„} is
given by
® = |p 2 — (l + q2j | [p(G2n+i — Gi) + (1 + q)(qG0 + G2n+2 — pG2n+i)
- p a t2i_1 - (i + q) i t t21]>i= l i= 0
(b) The sum of the odd indexed term s of {Gn} is given by
O = ^p2 — ( 1 + q 2 ) j. [p^ 0 + P ^ 2 n + 2 — P 2 G 2n+ l + ( 1 + q ) ( G 2n+ l — G i )
- ( i + q J A E t ^ - A p f : * ” ].i= l i= 0
Proof. From the recurrence relation (3.6), we have
pG2n — Gin+l + QG2n-l ~ At2n 1.
Therefore,
P E G2i = E G2i+1 + q £ G2i_x - A £ t2i- xt=l i=l t=l i=l
n+1 n n= E G2i_! +g E G2i_! - A E t2'- 1.i—2 t=l i=1
Hence,
P E G2i — E G2i_i + G2n+i — Gi + G2n+i + q E G2n_i - A E t2*-1t=l i=l i=l i=l
P T . G2j — (1 + g) T . G2i_i + G2w+i — Gi — A t2t *. (3.22)i=l i=l i=l
We write above equation as
pE = (1 + q)0 + G2n+i — Gi — Ea - (3.23)
Also,
p ]C GWi — G2i+2+ g y^ (*2i — A y) t2t2=0 i=0 i=0 i=0
n+1 n n= y ^2 i + q y ) 2*—a y t2t
i=1 2=0 2=0
= ^2 (*2 i + G2n+2 + <? y G2i + qGo — A y tt=l i=1 2=0
2i
(1 + g) X) 2* + G2n+2 + qGo — A Y t 2\2=1 i=0
Therefore,n+1 n n
P Z) ^ 21+1 — (1 + g) X) <J2i + G2n+2 + qG0 — A j>2 t2t2=1 2=1 i=0
p Z) G2i-i + pG2n+\ — (1 + g) Z) G2i + G2n+2 + qGo — A Y11212=1 i = l i= 0
i.e.,
pO — (1 + q)E — pG2n+i + G2n+2 + qGo — A y t2%. (3.24)2=0
Solving equations (3.23 ) and (3.24) simultaneously , we get
E = {p2 _ ( l +<72)} \p{G2n+\ — G\) + (1 + q)(qGo + ^ 2n+2 * pG2n+i)
p A Y ,? ^ 1 ~ (1 + q )J2 t2i]i=l i=0
and
O = -r-z— j——^-r\pqGo + pG2n+2 — p2G2n+i + (1 + q)(G2n+i — G{){P2 ~ (1 + g2) }
- (1 + q )A ^2 t2i~1 - A p f2 t2% 2=1 2=0
□
Next we have the following identity.
55
Proposition 11. For m, n > 0, GmGn - qGm_iGn_i = (b - zt)Gm+n_i
+(z - a)qGm+n_2 + (aq - bt)ztm+n"2 + z[tnGm - tmGn - q t ^ G ^ x - q U ^ g ^ ] .
Proof. Using Binet type formula (3.12),
L.H.S. = (Clam + c2/T + ztm)(Clan + c2/3n + ztn)
- q{c\am~1 + ca/T- 1 + zt™-1) ^ 1an~1 + c^ " ' 1 + z t ^ 1)
= (cjam+n + 4 fim+n + zHm+n + ci c2(ampn + fiman) + clZ(antm + amtn)
+ c2z(pntm + /3mtn) ~ q[{c\am+n- 2 + 4/3m+n~2 + zHm+n~2)
+ cic2{am~1 + c1z(an- 1tm~1 +
+ c22(/3n- 1tm- 1 + /sm- 1r - 1)]
= Ciam+n~2(a2—q)+c%/3m+n~2(/32—q)+z2tm+n~2(t2—q)+CiC2a'm~1l3n~1(af3—q)
+ c1c2F n- 1an-\aiP - q ) + ztnGm - z2tm+n + ztmGn - z2tm+n
- q{ztm~^Gn—i - z2tm+n~2 + ztm" 1 - z2tm+n- 2}
= Ciam+n~2Ci(ad) + c2 m+n~2c2(/3d) - z2tm+n + ztnGm + ztmGn
- q{ztm~lGn- 1 + zF-'G m -i - z2tm+n~2}
Since, a — /3 = d, c\d — (b — a/J) — z(t — 0) and c2d = (aa — b) — z(a — t ),
L.H.S. = c1a m+n- 1[ ( & - ^ ) - z ( t - /0)]-c 2 m+n- 1[ ( a a - 6 ) - ^ ( a - t ) ] - ^ r +n
+ ztnGm + ztmGn - 92{ r - 1G„_1 + tn- 1Gm_1 - ztm+n~2}
56
= 6(c1a m+n-1+c2/3m+"- 1)—ag(cxam+n-2+c2/ ^ +n~2)—2t(c1o:”l+n-1+c2/3m+n'-1)
+ zq(ciam+n~2 + c2/3m+n- 2) - z2tm+n + z(tnGm + tmGn)
- q z i t ^ G n ^ + F-'Grn-X - Ztm+n~2}
= bGm+n-1 — aqGm+n- 2 — ztGm+n- 1 + zqGm+n- 2 — bztm+n~1 + aqztm+n~2
+ z(tnGm + tmGn) - q z i^ - 'G n -! + F -'G m ^}.
= (b - ztfGm+n-! + ( z - a)qGm+n- 2 + (aq - bt)ztm+n~2
+ z[tnGm - tmGn - qtn~1Gm-i - q r- 'g n ^} .
□
With m — n in above , we get
Corollary 1 .
Gn - qGn-i = (b - zt)G2„-i + q(z - a)G2„ - 2 + (aq - bt)zt2n_2
+2z[tnG„ - qtn_1Gn_i].
Following identity is another version Catalan type identity. .
Proposition 1 2 . G„+rGn_r — G 2 = eqn_ru2_x + z tn[trGn_r + t _rGn+r — 2Gn].
Proof. L.H.S.=(Clon+r + c2/3n+r + ztn+r)(cian~r + c2/3n~r + ztn~r) - (Clan +
c2/3n + z tn)2
=cfa2n + 4/32n + z2t2n + ci c2(an+rPn~r + a n~r/?n+r + ztn+r{Clan~r + c2(3n~r) +
ztn- r(c1an+r+c2/3n+r) - { 4 a 2n+c2l32n+z2t2n+2(c1c2anl3n+zc2pntn+zc1tnan)}
57
=c1c2anpn(arp -r+a-rpr-2)+ ztn+rGn„r- z 2t2n+ztn- rGn+r-zH 2n-2 ztnGn+
2z2t2n
=clc2{ap)n~1'{ar - pr)2 + ztn[(trGn—r + t~rGn+r ~ 2Gn](ar - ffr)2
=eqU~' ( a - p y + ztn^ trGn-r + t~rGn+r - 2Gn]
—eqn~rU2_1 + ztn[{trGn- r + t~rGn+r — 2Gn), where Un as mentioned in 2.5.1.
=R.H.S. □
Hence we have the following result.
Proposition 13. Gn_iGn+i - G2 = eqn_1 + ztn[tGn_j + t - 1 Gn+i - 2Gn].
Proof. Let r = 1 in proposition 12 above.. □
An expression for any even indexed term of {Gn} is given below.
Proposition 14. G2„ = ( -q )n £ (? )(-q )n_iGn-i - z[(pt - q)n - t2n].i=0 v ' u
Proof.
R.H.S. = (-q )n £ (— - z[(pt - q)n ~ t2n]
= {-q)n £ (—- ) "- i (cia;n_i + CtP"-* + z tr-1) - z[(pt - q)n ~ t2n]
= * t („ n_ 4) +* £ (n ) (rfM-*)'
+ * £ f ” .) {p tT ^i-q T - z[(pt - q)n - t2n]t=o \ n ~ v
= C i(pa - q)n + c2(pp - q)n + z(pt - q)n - z[(pt - q)n - t2n}.
Since, a2 = pa - q and p2 = pp - q, we get
58
R.H.S.= cia2n + c2/32n + zf2ri
= G2n.
Hence the result. □
We now verify above results by giving some examples.
3.5 Exam ples
In this section some of the above results are verified. We give particular values
to p, q, A , t and verify these identities. Consider the equation
Gn+2 = Gn+1 — 2Gn + (—1)”, with Go = 0, G\ = 1
Here, p = 1, q = 2, t = —1, A = 1.
First few terms of Gn are
Go = 0, Gi = 1, G2 = 2, G3 = —1, G4 = —4, G5 = —3, Gq = 6, G7 = 11, G$ — 0
For n — 5,
Example l.We prove (a) of proposition 9.
We have, 5i = — 1 , S2 — 6, Pi = —128, P2 = —6, vx + qV2 —
R.H.S.= E G2 = - 128 + 2( - 1)(3) + 6(2 - 1) + (~ 12°) =
and
8 then -248
»=o -8 -8= 31
L.H.S.= E G2k = Gl + G\ + ... + G| = 31.fc= 0
L.H.S.=R.H.S Here (a) of proposition 9 is verified.
Exam ple 2. We verify (b) of proposition 9.
Here we take n = 6.
and Pi = —19, P2 = —241, Si = —7, S2 — 7.
59
Therefore,
L . H . S = 2 GkGk+i — GoGi + G 1G2 + . . . + G 6 G 7 = 6 4fc= 0
and
RII £ ~ 19 + (~ 2) ~ ^~241^ ~ 4) + 2(~7)(1Q) ~ (7)(~6)-16
L.H.S.=R.H.S. hence part (b) of proposition 9 is verified.
Example 3. We prove proposition 11.
To verify proposition 2.5, we take m = 2, and n = 3.
L.H.S.=G2G3 - 2GiG2= 2( - l ) - 2(2) = - 6.
R.H.S.=-5 - 1/2 - 1/4 - 3/4 + 2/4 = - 6.
L.H.S.=R.H.S hence proposition 11 is verified.
Example 4 We verify proposition 12.
we take n = 6, and r — 2
z = \ ,e = - 2 ,n x = 1
L.H.S=G8G4 - Gg = -36.
R.H.S=—32 - 1/4(4 + 12) = -32 - 4 = -36.
L.H.S.=R.H.S, hence proposition 12 is verified.
Example 5. We verify proposition 14.
We take n — 5 .
L.H.S.= G2n = -22
R.H.S.=(—<?)" E ft)(-p /? )B_<Gn-i - z[{pt - q)n - t2n]
=-32(83/32) - l/4 (—244) = -83 + 61 = -22.
L.H.S.=R.H.S. hence proposition 14 is verified.
60
3.6 Properties o f {Gn} using Matrices
The matrix method used here permits improved computational convenience.
With the use of these tools, very important identities such as Casini identity,
Catalan’s identity, d’Ocagnes identity for {£?„} are obtained.
Theorem 15. If Mn —G7i- i Gn
Gn Gn + 1
where Gi s are generalised pseudo
Fibonacci numbers then |Mn| = eqn_1 + ztn(Gn_1t + Gn+it _1 - 2Gn),
where e = abp — b2 - a2q — z{bp - 2bt - 2aq + atp + A}.
Proof. We use the principle of mathematical induction on n. For n = 1,
L.H.S. = Mi |
Go G\
Gi G2
= G0 G2 — g \
— a(pb — qa + A) — b2.
Also
R.H.S. — e + z(at2 +pb — qa + A — 2bt)
= apb — b2 — a2q + aqz + az(t2 —pt + q)
— apb — qa2 — b2 + aA.
Here we use the fact that A — z(t2 — pt + q).
Therefore, L.H.S.=R.H.S. showing that result is true for n = 1.
61
Assume that the result is true for any positive integer k i.e.,
\Mk\ = eqk~x + ztk{Gk. xt + Gk+Xr x - 2Gk). (3.25)
We prove that the result is true for n = k + 1 i.e.,
\Mk+1 | = eqk + ztk+1{Gkt + G k^t*1 — 2Gk+x).
From equation (3.25)
= eqk~l + ztk(Gk- it + — 2 Gk).
Following the row operations on the determinant, we get
Gk- l Gk
Gk Gk+i
Ri(-q)—qGk~ i —qGk
Gk Gk+1
— — eqk — ztkq(Gk- \ t + Gk+it 1 — 2 Gk).
The operation R\ + pR.2 — gives
pGk — qGk-1 pGk+\ — qGk
Gk Gk+1
—eqk — ztkq(Gk- \ t + Gk+\t 1 — 2 Gk).
By swapping the rows of above determinant and writing
pGk+1 - qGk = Gk+2 - Atn, we get
Gk Gk+1
Gk+i — A tk~x Gk+2- A t keqk + zfiq{Gk- i t + G ^ r 1 - 2Gk) (3.26)
62
Now, consider the determinant
Gk Gk+1
Gk+1 - Atk~l Gk+2 — Atk
= (GkGk+2 — G2k+1) + Atk{Gk+it~1 — Gk)
— |-Wfc+i| + Atk{Gk+\t~1 — Gk)- (3.27)
From equations (3.26) and (3.27) , we get
|Mis+i| + Atk(Gk+\t 1 — Gk) — eqk + ztkq{Gk-\t + Gk+it_1 — 2 Gk)-
On substituting A = z(t2 —pt + q) in above equation we get
[Mfe+x| = eqk + ztk(qtGk-\ + qGk+it-1 -2qG k)~ z(t2 - p t + q)tk{Gk+\t~1 - G k)
= eqk + z[tk+1(qtGk-1 - Gk+1 - pGk) + tk(qGk - qGk + pGk+i)J
+ z[tk~1(qGk+1 — qGk-i-i) + Gktk+2}
Hence
\Mm \ = eqk + z[tk+1(-G k+x - (Gk+1 - Atk-')) + tk(Gk+2 - Atk) + zGktk+2\
— eqk + z[—2tk+1Gk+\ — tkGk+2 + tk+2Gk]
— eqk + ztk+1 \Gkt + Gk+2t 1 — 2G'fc+ij .
Thus the result is true for n = k + 1. This completes the proof. □
Theorem 16. Let Bk =Gn+k Gn
Gn+k+1 Gn+1
where G i’s are generalised pseudo Fibonacci numbers. I f dk = \Bk\, for k > 1
then
63
(i) dk satisfies the non-homogeneous recurrence relation
dk+2 = pdk+1 — qdk + AD(n,t)tk, where D(n,t) = Gn+itn — Gntn+1.
(ii) dk and generalised Fibonacci sequence Uk are related by
dk = ~eqnUk-i - ztn\tGn- k + tkGn+1 - tk+1Gn - Gn+k+i]
where e — abp — b2 — a2q — z{bp — 2bt — 2aq + atp + A}.
Proof, (i) We prove that
pdk+i - qdk = dk+2 ~ AD(n, t)tk.
L.H.S of equation (3.28)
=pdk+i — qdk
—p\Bk+i \ - q \B k\
Gn+k+1 Gn Gn+k Gn=p - q
Gn+k+2 Gn+1 Gn+k+l Gn+1
— p{G n+k+\G n+\ GnGn+k+2 q{G n+kG n+l GnGn+k+1)
—Gn+\{pGn+k+\ qGn+k) Gn(pGn+k2 qGn+k+1)
=Gn+l (Gn+k+2 - Atn+k - Gn(Gn+k+3 - Atn+k+1)
—Gn+iGn+k+2 - GnGn+k+3 - Gn+1Atn+k + GnAtn+k+1
Gn+k+2 Gn Gn Atn+k+
Gn+k+3 Gn+1 Gn+1 An+k+1
Gn Atn+k=|-Bn+2| +
Gn+i A n+k+1.
=dk+2 ~ A D (n, t)tk.
=R.H.S. of (3.28) as required.
Hence the result.
(3.28)
□
64
Proof, (ii) For k = 0, |d0| = 0.
For k — 1, using theorem 15, we write
|8 i| = —e — ztn\tGn~ i + tGn+i — t2Gn — G2].
Assume that the result is true for n = k.
We have to prove that result is true for n = k + 1., That is to prove
|£?*+i| = —eqnUk - ztn[tGn-k+i + tk+1Gn+1 - tk+2Gn - Gn+k+2].
From result (i) and theorem 15, we get
I-8 *4-1 1 — p\Bk\ - q\Bk-i \ + AD (n,t)tk~l
= p(~eqnUk - 1 - ztn[tG„-k + tKGn+x - tk+1Gn - G'n+A:+i])
— q(—eqnUk- 2 ~ ztn[tGn- k + tK 1Grn+i — tkGn ~ Gn+k\ + AD(n,
= ~eqn(pUk - 1 - qUk - 2 ~ ztn[t(pGn_k ~ gGn-fe-i) + tk~l(ptGn+i - qGn+1)
— tk(tpGn + qGn) — (pGn+fc+i + qGn+k] + Atk~1{Gn+\tn — Gntn+l)
= —eqnUk - ztn[t(pGn- k+i - Gn+k+2) - Atn~k + Atn+k - tk+1Gn+1
- tk+2Gn + Atn~k + Atn+k]
= —eqnUk - ztn [tGn-k+i + tk+1Gn+1 - tk+2Gn - Gn+fe+2] .
This completes the proof. □
It is interesting to note that d’Ocagne identity for {Gn} can be obtain by
taking m = n + k as follows:
GmGn+1 - Gm+iGn = eqnUk + ztn \tGn- k+l + tk+lGn+1 - tk+2Gm. k - Gm+2] •
65
Theorem 17. Let Ca =Gn Gn
GnTn+s Gn—r+s.where G i’s are generalized pseudo Fibonacci numbers. I f ds — \CS\ for s > 0
then (i) ds satisfies non-homogeneous recurrence relation
da+2 = pda+i - qda + AT(n, t)tn, where T(n, t) = (Gnts~r - Gn- rta).
(ii) da and generalised Fibonacci Sequence Ur are related by
ds = eqn~rUr-iUs-i + ztn[ts~rGn - tsGn- r + Gn. r+S - t~rGn+s],
where e = abp — b2 — a2q — z{bp — 2bt — 2aq + atp + A}.
Proof, (i) R.H.S.=pds+i — qda + A T(n,t)tn
=p(GnGn- r+a+1 — Gn-rGn+a+i) — q(GnGn-r+a — Gn- rGn+s) + Atn{Gnts~r —
Gn- rta)
—GnipGn-r+s+l ~ qGn-r+s) — Gn- r(pGn+a+\ — qGn+a) + A tn(Gnts~r — Gn- rta)
=Gn(Gn-r+s+2 - A tn+a~r) - Gn-r(G r+a+2 - A tn+a) + A tn(Gnta~r - Gn- rt s)
—da+2-
Therefore,
da+2 = pda+\ — qda + AT(n, t)tn. D
Proof, (ii) We prove the result by method of induction on s.
For s = 1,
|Ci| = eqn~rUr-i + ztn[tl~TGn - tGn-r + Gn-r+1 - t~rGn+i].
Assume that result is true for n = s. i.e.,
\Ca\ — eqn~rUr~iUa- i + ztn\ta rGn — tsGn- r + Gn- r+a — t Gn+S].
We prove that it is true for n = s + 1.
66
From result (i) and above equation, we get
l^+iI = P\c s\ - g|Cs_i| + A(T(n - l ,*)*"-1
|C.+ll = p(eq"-rUr. 1U .^ + z f [ f - rG„ - t‘G„-r + G„_r+, - r 'G n+,]
- q(eq"-rUr. lU ..2 + 2t“[ f - ' - ‘G„ - r~ 'G „.r + G ^ . , , - r-'G„+»_1]
+ A[G„-1t‘- r~1 — Gn — ft*"1]
= eUr - l(pf/s_i - qUs_2) + ztn[Gnts- r~1(pt - q) - t s" 1Gn_r(pt - q)
+ Gn-r+a-l - ^ n -r+ a-i _ tr(Gn+a+1 - A tn+S~2)}
+ A[Gn - It*-1- 1 - Gn-rt8- 1}
= eUr-iU, + ztn[t*-r+'Gn - t3+1Gn-r + Gn- r+s+1 - t~rGn+a+1].
Thus the result is true for n = s + 1. Hence the result. □
Note that if we take s = m — n + r in part(ii) above , we get generalized
form of d’Ocagne’s and Catalan’s identities. Further, if
1) m = n in part (ii) above, we get Catalan’s identity .
2) replacing n by n + 1 and r = 1 in part(ii) above, we get d’Ocagne’s identity.
3.7 A n oth er G eneralization
In this section we apply yet another generalisation of Fibonacci sequence to
{5n}. In 1967, Elmore [3] used the following idea to extend the Fibonacci
sequence. Here the exponential generating function E q{x ) is used.
67
gQi®_e^lXLet Eq{x) = — , a x, /3X being the roots of x2 — x — 1 = 0.
Define successively E ^x), E2(x) , ..., Em(x) by
El{x) = E'Q{x) = ^ X- ^v5
e 2{x) = e : {x) = ^ x - / ^ \v5
and so on. In general,
Em( x) = E t \ x )ameaix _ pme0ix
7Em > 1 .
Observe that Em+1(x) = £ m(a:) + Em^ (x ) .
Thus Em is another extension of Fn.
Taking a, ft as the roots of x2 - px + q = 0 and using a similar method as
above, we define
* : (* )=aneax - Pne0x
d (3.29)
where d = a — (j.
We need above function in some identities of {G„} in section (3.10).
Applying the similar process to {Gn} a new sequence {E*} is obtained as
follows:
Consider
E*0{x) = E*{x) = c\eax + aeP* + ztf*
as the exponential generating function of {Gn}. Further, let En(x) of the
sequence {E*(x)} be defined as the nth derivative with respect to x of E0(x),
then
E*n(x) = ci aneax + c2(3ne?x + ztnext. (3.30)
68
Note that
En(0) = cj a" + c2(3n + ztn
— Gnwhich in turn, reduces to Fn if A = 0, a = 0 and b = 1 .
(3.31)
Theorem 18. T he sequence {E*(x)} satisfies th e non-homogeneous
recurrence relation
E;+2 (x) = PE* + 1 (x) - qE* (x) + A tnext. (3.32)
Proof. Since a and /3 are the roots of x2 — px + q = 0,
pat — q = a2 , p(i — q = /32 and z{t2 - p t + q) = A .
R.H.S. = p(cian+1eax + c2pn+1epx + ptn+1ext) - q{cxaneax + c2pne0x + ztnext) + Atnext
= cianeax(pa - q ) + c2/3nePx(pj3 - q) + z tnext(pt - q) + z(t2 - p t + q)tnext.
= Ci a n+2eQI + c2fin+2e0x + ptn+2ext
= E*+2(x).
□
3.8 U se o f G eneralized Circular Functions
In this section, we discuss the generalized circular function which shall be used
for developing another generalization of the sequence {Gn}-
The generalized circular functions are defined by Mikusinsky [12] as follows:
Letfnr+j
t£ i(n r + j)Vj = 0, 1 ,..., r — 1 ; r > 1 , (3.33)
69
Mrj — y~!( —1 )■f-nr+j
n=o (n r+ j)V 3 = 0 , 1 , r — 1 ; r > 1 .
Observe that
(3.34)
Ni,o{t) e , N2<o(t) — cosht, N21 (t) = sinht and
Miflit) e , M2,o(t) = cost, M2,i (£) = sint.
Differentiating (3.33) term by term it can be easily established that
Nr,j-p{t), 0 < P < j,
Nr,r+j-p(t), 0 < j < p < r.
In particular, note that from (3.35)
N $ ( t) = AW t),
(3.35)
in general,
N § r\ t ) = Nrt0(t) ,r> 1 . (3.36)
Further note that
ATr,0(0) = N % \0 ) = 1 .
Using generalize circular functions and Pethe-Phadte techniques [19], an
other generalisation of sequence {Gn} is defined as below:
Let
Hq{x ) = ClNr,0(a*x) 4- c2Nrfl(P*x) + zN rto(t*x) (3.37)
where a*=a1/r, = P1/r and t* = t1/r, r being a positive integer.
Now we define the sequence {Hn(x)} successively as follows:
Hi(x) = H^r)(x), H2(x ) = H t \ x ) and in general
70
Hn{x)—Ho \ x ), where derivatives are with respect to x.
Then from (3.36) and using (3.37) we get
Hi(x) = cia Nr,0(a*x) + c2(3Nrfi(/3*x) + ztN rfl(t*x)
H2{x) = cxa2Nrv(oc*x) + c2j32Nrfi{p*x) + zt2Nrfi(t*x)
and in general,
Hn(x) = cxcxn Nrt0(a*x) + c2/3nNrfi(/3*x) + z tnNrfi(t*x). (3.38)
Observe that if r — 1, x = 0, A = 0, a = 0 and 6 = 1 , {i7n(:c)} reduces to
{Fn}-
Theorem 19. T he sequence {Hn(x)} satisfies th e non-homogeneous
recurrence re la tion Hn+2(x) = p H n+i(x) — qH n(x) + A tnNr 0(t*x).
Proof.
R.H.S. = p [Cla n+1iVTlo(a*x) + c2(3n+1Nrfl((3*x) + ptn+1Nrfi(t*x)]
- q [ClanNTt0(a*x) + c2/3nNr,Q(0*x) + ptnNrfi(t*x)} + AtnNrfl(t*x)
= C\anNTfl(a*x)(pa - q) + c2(3nN rfi(/3*x)(p/3 - q) + z tnNr,0(t*x)(pt - q)
+ AtnNr<0(t*x).
Using the fact that a and /? are the roots of £ — px + 9 — 0
and A = z(t2 - p t + q), we get
R.H.S=Cla n+2iVr,o(a*a;) + c2/3n+2N rfi(P*x ) + ztn+2Nr,0(t*x)
□= Hn+2(x).
71
Observe that for r - 1 , a* = a, {3* = p and Nrfi{t)= N ^ t ^ e * equation
(3.38) becomes
Hn{x) = Claneax + c2pne0x + Atnetx (3.39)
= K ix ) .
In addition to above particular value of r and x = 0, E*(x) reduces to Gn.
Further with the above values of r ,x ,p = 1 , q = - 1 , a = 0, b = 1 and
A = 0, Gn becomes Fn. i.e.,
Hn(Q) = En{ 0) = Fn.
3.9 Som e Id en tities o f Hn(x)
In this section some identities of the sequence {Hn(x)} are proposed and
proved. The following identities corresponds to identities (4.5) to (4.9) in
the Walton and Horadam paper [28]. These identities can be easily proved by
using Binet type formula (3.12).
In the following, let Na = Nrfi(a*x) with the corresponding expression for Np
and Nt. Also let <5„,r(ci,a;x) = cianNrfi(a*x) with the corresponding expres
sion for Q„,r(c2, /?; x) and Qn,r(z, x).
Note that Hn(x) = Q„,r(c 1,a ;x) + Qn,r (c2,/3;x) 4- Qn,r(z,t-,x). We have the
following identities:
(i) Hn+1 (x)H„_i(x) - H n(x ) 2 = eqn_1NQN^ + z tnN t{Hn_!t + Hn+it" 1 - 2Hn}
72
Proof.
L.H.S. = (cian+1Na + c2pn+1Np + ztn+1 N ^ a 71- 1 Na + c ^ 1 N0
+ztn~1Nt) - (cia nNa + c2pnN0 + ztnNt)2
= clc2ocn+1pn- lNaNf} + ci ztn~lan+1NaNt + c1c2an~1 n+1NaN()+
c2ztn~1(3n+1NpNt + c1zan~1tn+1NaNt + c2ztn+l pn~x N0Nt
- 2 (Clc2anpnNaNp + Ciztnan NaNt + c2ztn pn N0Nt)
= c1c2anpnNaN0(a//3 + P /a - 2) + z t ^ 1 Nt(Clan+1 Na + c2pn+1N0)
+ztn+lnt(cian~1 Na + c2pn~lNp) - 2ztnNt(ClanNa + c2pnNp)
= cic2NaNp(aP)n~ \a - p f + ztn~x NtHn+xztn+l NtHn. x - 2ztnNtHn
=eqn~1NaNp + ztnNt{Hn^ t + H ^ r 1 - 2Hn}. =R.H.S. □
(ii) H„(x)F*+1 (x) - qH n_1 (x)F*(x) =
eQXQ„+8+ i(ci,a ;x ) + e^xQn+s+i(c2, A x) - Q n(z,t;x)[F*+1 (x)t - F ;(x )t_1]
Proof. Using equation(3.29) and (3.38), we get
L.H.S.=(ci anNa + c2pnNp + ztnNt){------- ---------------)S g Q X _ Q S e 0 X
-q{(c ian~lNa + c2Pn~lNp + ztn~lNt){-----q -_ ^ -----)}
= i{ c ia n+s+1eQI7VQ + c2as+l PneaxNf) + ztnas+1eaxNt} a
— i {ci ane0xNQPs+1 + c2pn+s+1e0xN0 + ztnPs+1e f} a
{cian+s~1eaxNa + c2ots Pn~l N0eax + ztn~xNtotseax} a
73
^{cian 1e0xNaPs + c2p n+a~xNpeP* + ztn~lNtpse0x}
=2 (Cl a n+seQI ATa (a - q/a) +c2 of p p - ^ N p (a p -q )+ ztnaseaxNt (a -q /t)
+2 {cian~le0xNapa(q ~ aP) + c2pn+s~1e0xNp(q ~ p2) + ztn~x Ntpse0x(q -
m
= h c 1an+s+1eaxNa(d)+c2pn+3+1e0xN0(d))+ztnNt^~~aS+1r e^ +1) d d
- 7tn - ix (<*seax - Ps^ x)d
=eaxQn+s+1{ci,a\ x)+e0xQn+s+i (c2, 0; x )-Q n(z, t; x ^ F ^ ^ t - F ^ x ) ^ 1].
=R.H.S. □
(Hi) Hn(u)F;+1 (v) - qHn_i(u)F*(v) =
eQVQn_i,r(ci, a; u) + e^vQn_i,r (c2,P; u) + Qn-i,r(z, t; u)[F*+1 (v) - q/tF*(v)]
Proof. L . H . S + c2p nNrfi(p*x) + z tnNr,0(t*x:))[“* c” l f
—q(ciCen~* Na + c2pn- lN0 + ^ ”- 1iVr,o ( ra ;) ) [^ E f^ ]
[c1a n+s+1iVQeQt' + c2Npeav Pno.s+1 + ztniV'rto(t*a:)oJ+1em’a — P
—C\ocnNaPs+1e0v - c2NfiPn+s+1eftv - z tnNrfi(t*x)ps+ie0v]
a —— [c\a n+s+1Naeav - qc2N 0easpn- x - qztn- 1Nr,0{t*x)asea«
-C io rN .F + 'e* - - rfW ,,„(r*)/J '+Ie'>'1
74
= ^ g ^ ian+SN“eaV(a ~ */<*) + c iN p e rp o fic t - q/l3)
+ ztnNri0(t*x)aseav(a - q/t) - c1an0sNae^v{0 - q/a)
- c2N ^ v0n+a{0 - q/0) - z tnNrfi(t*x)0se^v(0 - q/t)}
= ^ — [ c ia ^ '^ c ^ C a - 0 ) + c2Npeav0na s(a - q/0)
+ ztnNTfi( fx ) a aeav(a - q/t) - Clan0sN ae0v(0 - q/a)
+ c2Npe ^ 0 n+a(a - 0) - ztnNrfi(t*x)0se0v{0 ~ q/t)}
= Clan+aNaeav + c2Nf)e0v0n+a + ztnNr<0(t*x)E*s+1(v)
— ztn~1Nrfi{t*x)qE*{v)
= C\Naeavotn+a + c2Nfiefiv0n+a + ztnNrfi{t*x) {E*s+1(y) - qE*(v)/t}
= e^Qn+aAcucr, u)+effvQn+s>r(c2,l3;u)+Qnir(z,t;u) {Fs*+1(u) - qF*{y)/t}.
=R.H.S. □
(iv) H2 (x) - qH’ .^ x ) = dQ5>r(Cl>a;x) - fQ 2 ,r(c2 )/3;x)
+Qn>r(z,t;x)[2H n - 2 qHn_x/t] - Q’ ,r(z ,t;x )[l - q /t2]
Proof. L .H .S .= (c ia niVQ + c20 nN 0 + z tnNrfi{t*x))2
- q{cian~l Na + c20n~l Np + z tn- 1N r,0(t*®))2
= (c 2a 2niVr20(a * x ) + 4 0 2nN/fi{0*x) + z2t2nNr,o(t*x )
+ 2 cxc2an0nNaN0 + 2clza ntnNaNr,o(t*x )
+ 2c2z0 ntnNf}NTSi(t*x)) - q {4»2n' 2Nrfi(a*x)
+ c2/?2" - 2^ 2 o(/?*x) + zH2n- 2N rfi( t'x ) + 2c1c2a n- 10n- 1NaNp
75
+ 2cizan~1tn~1NaNrfi(t*x) + 2c2z0n- 1tn- 1N0Nrfi{t*x))
= 4a2nN 2 0(a*x)(l - q /a2) + 4/32nN 2o{0*x)(l - q /02)
+ 2cic2an0 nNaN0{\ - q/(at0)) + 2clZantnNaNr^ t * x)(l - qf(at))
+ 2zc2tn0nNrfi{t*x)N0(l - qj(t/3))
Since a@ = q, we get
L.H.S.=c?a2n- 17Vr2io(o;*x)(Q! - 0) + <%0*n~1N*fi(0*x){0 - a)
+ z2<2nAr20( rx )( l - q/t2) + 2ztnNrfi(t*x){ClanNrfi(t*x) + c20nNrfi(t*x))
- 2zqtn~1Nr<0(t*x)(ciOtn~lNr$(t*x) + c ^ - 'N p )
=c2a 2n- 1AT20(Q*x)(a - P) + 4 P 2n- 1N 20{/3*x)(/3-a) + z2t2nN 20(t*x)(l -
q/t2) 4- 2ztnNr<o(t*x)Hn - 2z2t2nN 20(t*x) - 2zqtn~lNr {t*x)Hn-i
+ 2qz2t2n~2N 20(t*x) - qztn- lNrv{t*x)Hn^
=4o/2,n~1N 2o(a*x)(a - 0 ) - 4/32n~ 'N 20(l3*x)(a - 0 ) + z2t2nN 2fi(t*x)( 1 -
q/t2) + 2zNrfl(t*x)tnHn - qtn~^Hn—i
=^Qn,r(ci»<*; x )- jjQ l r(c2,0\ x)+Qntr{z, t ;x)[2Hn-2qH n-x /t] -Q ltr(z, t; x )[l-
q/t2}
=R.H.S. D
(v) H5+1(x) - q’H ^ f x ) = # Q ; ,r(c ,,a ;x ) - ^Q S,t(c2,/3;x)
+ S Q i.r(2 .t;x ) + Q »,r(M ;x)[H n+1 - q’ l t - i t - 1].
Proof.
L.H.S. = (Clan+1Na + c20n+1N0 + z tn+1Nt)2 - q2{cla n' 1Na + c20n^ N 0 + ztn xNt)2
76
(£o?n+2N 2a + & 2n+2N 2 + zH2n+2N 2 + 2c1c2(a/3)n+1 NaNp + 2ztn+1 Nt(Clan+1 Na
+c2/?n+1iV ) - q2(c2a2n~2N 2 + c2l32n- 2N% + z2t2n~2N 2 - 2q2c1c2{a/3)n- 1NaNf)
- 2 q2ztn- 1Nt{c1an- 1Na + c ^ N p )
= ^ a 2nN l{a2 ~ Q2/<*2) + & 2nNj((32 - q2/(32) + z2t2nN 2(t2 - q2f t2)
+2ztnNtHn+i - z2t2n+2N 2 - 2q2ztn~lNtHn^ + q2z2t2n- 2N 2
= (?xa2nN 2(q2 ~2- 2- - 4(32nN 2t a- j J Q + 2ztnNt{Hn+1 -
+2 q2z2t2n~2N 2
= ^ Q n , r ( Cl > X ) ~ ^ Q n , r ( C2 ) / 3 ; ^ ) + ^ Q ^ r( z , t ‘,x ) + Q n,r{z, t \x)[H n + 1pd,
-q 2Hn-.it"
□
77
Chapter 4
Pseudo Tribonacci Sequence
4.1 Introduction
One of the well known generalization of Fibonacci sequence is the Tribonacci
sequence. Tribonacci sequence is defined by a third order homogeneous recur
rence relation. It has been studied extensively and its various properties are
found in [13],[14], [22] and [23]. In Chapter 3 a new extension of Fibonacci
sequence called pseudo Fibonacci sequence has been introduced using non ho
mogeneous recurrence relation. We now extend this concepts to Tribonacci
sequence. Here third order non homogeneous recurrence relation is considered
to derive a new sequence called pseudo Tribonacci sequence. We obtain some
standard identities for this sequence. Examples are given in support of some
identities. Further using E operator, we show that this sequence is reduced
to second order generalized pseudo Fibonacci sequence. Pseudo Tribonacci
sequence is further generalized using circular functions and Pethe - Phadte
technique reported in section (2.5.9). We define the following :
Definition: The pseudo Tribonacci sequence {Jn} is the sequence satisfying
78
the following non- homogeneous recurrence relation
J„ = pJ„_i + q J„_2 + r Jn_3 + Atn~3, for n > 3, t ^ a , /3,7 , (4.1)
where A ,t £ Z , p,q ,r are arbitrary integers and oc,/3, 7 are distinct roots
of auxiliary equation xz - px2 - qx - r = 0. Let the homogeneous relation
corresponding to equation (4.1) be
Hn = pHn- i + qHn- 2 + riJn_3, (4.2)
with the seed values
Ho = 0, ^ = 1, tf2 = p. (4.3)
A Solution J„ of the recurrence relation (4.1) is
Jn = Hn + 4 P)
where //„ is a solution of homogeneous equation(4.2) and is a particular
solution of (4.1).
From the characteristic equation we have
Hn = ci c*n + c2pn + c37n,
with
a + /0 + 7 = p, a /3 + P'1 + 7« = “ 9, otM = r. (4.4)
Using the initial conditions in (4.3), and solving, we obtain
<*(7-0) „ ~ TO = 7(/3 - « ) (4.5)ci = ---- £ ---- . c2 - £ ’ C3 A
79
where A = (7 - P)(y - a)(p - a).
Let the particular solution of (4.1) be J ^ = Dtn. Then we have
Dtn = pDtn~l + qDtn~2 + rD tn~3 -f A tn~3.
and hence
D = (t3 - pt2 - qt - ry (4-6)
Using (4.5)and (4.6),we get
Jn = Hn + 4 P)
= i { ( 7 - + (a - + (fi - ay r ' ) + w _ ^ _ ry
This is Binet Formula for the pseudo Tribonacci sequence.
Denote the sequence Hn by H^, and H 2, when the seed values are
H0 = 1 , Hi = 0, H2 = q (4.7)
and
Ho = 0, Hx = 0, H2 = r (4.8)
respectively. Denote the sequence {Jn} by { J*} and {J^} with the initial
conditions as in (4.7) and (4.8) respectively. Then we have,A tn
Ji = j{ ( 7 2 - 02)a"+' - (I2 - a2)? '* 1 + - aI)T”+1> + ( t s - p p - q t - r ) '
andA tn
Jl = i«7 - 0)0." + (a - 1)0" + (P - o)7"} + ((s_ pi2 -q t-T ) '
80
4.2 Som e id en tities o f {Jn}
In this section we shall obtain some usual identities for pseudo Tribonacci
sequence {Jn}.
Proposition 20. The generating function G(x) of Jn is given by. x (l - tx) - Ax3 , ,
G[x) = j-— —rj-------------- 5------- provided \tx\ < 1.(1 - tx){ 1 - p x - qx2 - rx3) ' 1
Proof.
Let G(x) = Jnxn. (4.9)n = 0
, oo , ooThen x~1G{x) = £ Jnxn 1 = £ Jn+ix". Hencen=0 n=—1
x-'G ix) = Y Jn+iXn + Jox '1. (4.10)n=0
SimilarlyOO
X~2G(x) = "*22 Jn+2Zn 4" Jox 2 4- J\X 1 (4.11)n=0
andOO
x ' 3G{x) = Y Jn+3Xn 4- JoX~3 4- JiX- 2 + J2®"1. (4.12)rt=0
Multiply corresponding equations (4.11), (4.10), (4.9) by p ,q ,r respectively
and subtracting them from equation (4.12), we get
G(x)[x-3 - px-2 - qx 1 - r] = Y .(J n +3 ~ pJn+2 — qJn+i rJn)x71=0
+ (J0x~3 + J lX '2 + ]2X~l) - (JoX~2 4- J lX '1) - JqX 1,
81
which yields,
G(x)[l - p x - qx2 — rx3] =OO
A E (tnXn+3) + J° + JlX + ~ P JoX + JlX2) ~ rJ°x2-Thus OO
x - A x3A £ (tx)nG(x) = ------------- n f ----- -
I — px — qxz — rx3
Hence
G(x) = x(l — tx) — A x3 (1 — tx)( 1 — px — qx2 — rx3) ’ provided |£a:| < 1.
□
The Exponential Generating Function E*{x) of Jn is given by
E*(x) = c\eax + c2ePx + c3e^x + zetx (4.13)
where Ci, c2, c3 and z are constants .
We now obtain the sum of the first n + 1 terms of Jn.
Proposition 21. The sum of first n + 1 terms of the sequence {Jn} is given
by
1 J2 + (1 — P)(Jo 4" Jl) — QJq (< n+2 "b Jn+l) H- pJn+1 rJn + A Sh k ~ (1 - p - q - r )
n—1where S = £ tk and p + q + r ^ 1.
*= 0
Proof. From the recurrence relation Jn+3 — pJn+2 + 9 Jn+i + rJn + taking
values for n, we get the following:
For n = 0, J3 = pJ-i + 9«/i + rJo + ^ ° -
82
In general, for n ~ k, J* = pJk-i + qJk- % + r Jk~§ + A tk~3.
Adding these equations,term by term, we get
For n — 1, J4 — pj$ -j- q j2 + r j \ -(- A t1.
Hence,
' L Jk- J° - J i - J 2 = P ' E Jk - P ( J o + Ji + Jn) + q ' £ j k ~q( J0 + Jn_1 + Jn)fc=° fc=0 A:=0
n n _3
~t~r ^ — r (Jn- 2 + «/n- l + Jn) + A f*.*=° fc=0
On simplification,
n(1 - p - q - r) £ Jfc = (Jo + Ji + J2) - p ( J o + Ji + Jn) ~ q(Jo + Jn- 1 + J„)
Jt=071 — 3
r{Jn- 2 + Jn-l + Jn) + A tfcfe=0
= (J0 + Jj + J2) - pJo - pJi - (pJn + qJn- 1 + r J„_2)n-3
9 Jo ?Jn T Jn- l rJn "b A ^ tk=0
= ( Jo + Jl + J 2) — P Jo — P Jl ~ (Jn+ 1 — At"-2) — q Jo+P Jn + 1 — (p Jn+ 1 +9 Jn+T’ Jn-l)71—3
- r J n + A ]T tfck O
n—2= (Jo +Jl + J2) —P( Jo"b Jl) — Jn+ 1 ~ 9 Jo+pJn+l — ( Jn+2 — At ) —rJn + A ^^t
fc=0
n—1
= J2 + ( 1 — p)( Jo + Jl) ~ 9 Jo ~ (Jn+2 + Jn+l) + pJn+ 1 — rJn + A t*.
Therefore," J2 + (1 - p )(J o + J l) - q Jp ~ (Jn+2 + Jn+l) + pJn+1 ~ ^Jn + AffE Jit -------------------- -fc=0 (1 - p - q - r )
□
83
Now we state and prove some identities involving summation of product of
two terms of the sequence Jn. We use the following notations. Let
V\ = 1 + P2 - <72 - r 2, v2 = 1 - p2 + q2 - r2,
V3 = 1 + P2 + Q2 ~ r 2 , for any integers p, q and r such that p + q + r ^ l .
Si = t Jktk and S2 = t t2k.fc=0 fc=0
mi = (J2+ J 2 + J 2) - (J2+1 + J 2+2 + j 2+3) + p2( J 2 + J 2) _ p2( J 2+i + j 2+2) _
tfUo ~ ^n+i) — 2p {JqJi -h J1 J2 ) + 2p(Jn+i Jn + 2 + Jn+2 Jn+3 ) + 2A(r + qt~1)S\ —
2Aqt~l(JQ — Jn+itn+1) + A2S2,
m, = (J2 + J2 + J2) - (J l+1 + J l+2 + Jn\ 3) — p2(Jq + J 2) + p2{J2n+l + J 2+2) +
<?(Jo ~ J%+j) + 2[>l(r + pt~2)Si — q(JoJ2 — Jn+iJn+3) — Ap(J0t~2 + J it" 1 +
•fn+l *-1 — ^n+2 ”)] + S2,
mz = ( J g + J ? + J |) - ( ^ +1+J^+2+ J ^ 3) - p2( ^ +i + J ^ 2- J i2)+?2(J02- 4 2+i)+
2[p9(^0‘fl ~ Jn+lJn+2) +P(Jn+lJn+2 + J„+2Jn+3 ~ • O'Tl — ^1^2) + <l{Jn+\Jn+3 ~
JoJ2)+Aqt~1(Jo — Jn+itn+1)+Apt~2(Jo+ Jit—Jn+itn+1 — Jn+2tn+2)—At 3(Jo+
Jit + J2t2 - Jn+itn+1 - j n+2tn+2 - J„+3*n+3) - A{qt-1 + p r 2 - t"3)]S! - A252.
We have the following.
Proposition 2 2 . For any integer n > 0,
4(p + rq)\(q + rp)m 3 — <?m2)] — 4p(q + rp)( 1 — g)mi(V E JkJk+i — -------------------------- x 'k=0 0
n 2f (a + rv)viv3 — <7m2t>3 + qmiv2 — {q + rp)miu3]f t) E JkJk+2 = ------------------------ xfc=o 0
. , n _ 2 n d -Q)(v im2 - m i V 2) + 2(p + rq)(m 3V2-m 2V3)(Hi) Z J k = --------------------------------5
84
Ui - 2 (p + rq) 0
provided 5 = 0 — 2(q + rp)
0 - 2g
= 4(p + rg)[(g + rp)v3 - gujj)] - 4p(9 + rp)(l - q)Vl ± 0.
Proof. Using the recurrence relation (4.1), we write
(Jk+3 ~ pJk+2)2 = (QJk+1 + **«/* + A tk)2.
i.e. Jl+ 3 + p2 Jk+2 — 2pJk+2Jk+3
= W + I + r2 fc + A2t2k + 2(rg Jk Jk+1 + rAJktk)
+AqJk+ltk4+3 + p2Jk+2 ~ q2Jk+1 - r2 4
= 2\pJk+2Jk+3 + rqJkJk+i + rAJktk + AqJkJrltk + A2t2k}.
Taking summation from 0 to n on both sides, we get
t 4+3 + P* £ Jk+2~q2 £ Jt+x - r 2 £ 4fc=0 fc=0 fc=0 fc=0
= 2[p 52 Jk+2Jk+3 + rq 5Z JkJk+i + rA 52 Jktkfc=0 fc=0 fc=0
+Aq £ Jk+ltk + AH2k).k=0
Hence,
(1 + p 2 - q 2 - r2) £ 4k=0
= ( 4 + 4 + 4 ) ~ (4 + i + 4 +2 + ^n+3)
V ( J j + J f ) - p ’W + i + •£ « ) - ?2Oo - •£«>
+2 (p + rg) X) JkJk+i — 2 p( JriJ] + /i J-i)*=0
+2p(Jn+1 Jn+2 + J n+2 J„+3) + 2A(r + q t-1) 52 - 2A qt~ \J0 ~ - W " )
+ 4 2 £ f2*,fc=0
85
which is written as
ViX = 2 (p + rq)Y + mi (4.14)
where X = £ J%, and Y = E Jk Jk+1. fc=o fc=0
Similarly from equation (4.1), we write
( *+3 ~ 9«4+i )2 = {pJk+2 + rJk + A tk)2
i.e Jl+3 + q2 Jk+i ~ 2qJk+1Jk+3 = p2 Jfc+2 + r2 + A2t2fc + 2{rpJk Jk+2 + ArJktk +
ApJfc+2**}-
After arranging the square terms together and summing both sides up to n
terms, we get
,E 3 — P2 E Jk+ 2 ~ 92 E fc+i ~ r2 ~ 2{< 7 X) Jk+iJk+s + rpJkJk + 2 +t =0 fc=0 fc=0 fc=0 fe=0
ArJktk + ApJk+2tk} + A2 Y, t2k-k=0
Hence,
£ 4 - (jg + j* + j i ) + ( ^ +1 + ^ +2 + j**,) - p2 £ ^ + p2^ + j ?) -fc=0 fc=0
^ 1 + ■£«) + Q2t 4 - iHM - Jin) - r2 £ j?fc= 0 fc=0
= 2(7 53 JkJk+2 — 2q(JoJ2 Jn+lJn+z)~^~TP 53 ^n^n+2+-^r 53 Jktn+Apt 2 53 JkJk~~fc=0 fc=0 fc=0 fc=0
Apt~2{J0 + J ,t - J n+1t"+1 - Jn+2 in+2} + ^ 2 E t2fc-k—0
This yields,
( l - p 2 + 92 _ r2) £ J 2 = (J2 + J 2 + Jf) - ( J 2+1 + J 2+2 + Jn+3) ~ PV o + Jf) +k=0
f { J l +1 + +2) + 92( ^ -^ n + i ) + Eo JfcJ fc+2+ rP fcE ^fc+i^+2 + r fcE +
Apt~2 E «/fc£* + 2Apt~2(Jo + «/l£ — Jn+ltn+1 ~ Jn+2’tnJt2] + A E * • fc=0
= 2[(g + rp) E Jfc Jfc+2 + A(r + p t '2) E «/*** - Apt~2{J0 + Jit - J„+itn+1 -fc=0 fc=0
Jn+2r +2] + ( J 2 + J 2 + J?) - (J 2+l + Jn+2 + Jn+s) ~ P2(J0 + ^?) + ^Vn+X +
86
ft+%) + Q2(Jq - <%+i) + A 2 £ t 2k.fc=0
Writing the above equation as
v2X = 2 (q + rp)Z + m2, (4.15)
nwhere Z = £ JkJk+2 -
k=o
Also from the relation (4.1), we write
(Jk+3 ~ fJk)2 = {pJk+2 + qJk+l + A tk)2
i.e, Jfc+3 + r2Jfc — 2rJkJk+z — P2 Jk + 2 + Q2J%+i + ^{pqJk+iJk+ 2 + AqJk+itk +
ApJk+2tk} + AH2k.
Therefore
Jk+3~ r2Jk~P2Jk+2~(l2Jk+l ~ 2'[Jk+3(Jk+3~pJk+2~QJk+l~Atk)+pqJk+iJk+2+
AqJk+itk + A pJk+2tk] + A2t2fc.
= ‘ [^kJr3 ~ P J k + 2 J k + 3 ~ ~ (l J k + \ J k + 3 ~ A J k + z t k+ p q J k + l J k + 2 Jr A q J k + i t k+ A p J k + 2 t k]Jr
AH2k.
Therefore,
Jk+3 + p2Jk+2 + q2Jk+1 - r2*-b = 2[pJfe+2J*+3 + qJk+iJk+3 - pqJk+iJk+2 -
AqJk+itk - A p jk+2tk + A tkJk+3] - A 2t2k.
Summing up both sides, we get
i j Jw + p2 £ £ 4 + i - r 2 £ n = 2b | oa « a « + » r : a +^>,+3 -
w £ ^ e - w " - ^ £ JM t " + A £ *«<*] - ^ £,«“ •
=2[pE a j m + , E A - w E JkJk+1 -A q t - 1 f Jkt‘ - E A*‘ +*=2 fc=l fc=1
A r 3 E3 7fctfc] - A2 1 : t2k.k=3 fc=0
Therefore,
87
( l V + ^ - r 2) 1 J l = 2 p ( l - 9) ± JkJk+l+2„ £ Jkj M _ 2A(qt-' +pt~2 +«—0 fc=0
r 8) £ 4* fc - A2 £ *2fck=0 fc=0
+(jg + J2 + J|) + ( J 2+1 + j2+2 + J2+s) + p 2( J2 + j2) + g2( J2 _ j2+^ + JQ J% _
Jn+1*4 +2) + Aqt 1(J0 — Jn+1tn+1) — (J0 4- Jxt -f J2t2 _ Jn+1£n+l _ Jn+2tn+2 —
^n+3«n+3)
which can be written as
— 2p(l — g)K -+- + m3 (4.16)
We solve the equations (4.14),(4.15) and (4.16), to get
* = £ 4 4 + ifc=0
4(p + rq)[(q + rp)m3 - qm2)] - 4p(q + rp)( 1 - ^)mi5
y = £ .4 *4+2k=0
_ 2[(q + rp)viv3 - qm2V3 + qmiV2 — ( g + r p ) m i ^ 3]
5
and
fc= 0
2p(l - q){v\m2 - m \v2) + 2(p + rq)(m3v2 - m2 3)= ■ j ’
□
88
We illustrate the above result with an example.
Example Consider the equation Jn+3 = 2 Jn+2 + Jn+1 + Jn + ( - 1)".
Here p = 2, q = 1, r = 1, t = - 1 and A = 1.
First few terms of this sequence with usual seed values (4.3) are
Jo = 0, Ji = 1, J2 = 2, J3 = 6, J4 = 14, J5 - 37, J6 = 93 .
Computation of various required quantities yield, vx — 3, v2 = —3 and t>3 = 5.
For n = 2, m i = -69, m2 = -219, m3 = -43, si = 3, s2 = 1 and S = 216.
Proposition 22 (i)
L.H.S.= 5.
R.H.S.=1080
4(p + rq)[(q + rp)mz - qm2)] - 4p{q + rp)(l - g)m!
= 5.216Result is verified.
Proposition 22 (ii)
L.H.S.= 14.2[(q + rp)v\v$ - qm2V3 + gmin2 - (q + rp)mit>3]
R.H.S.=3024216 14'
Result is verified.
Proposition 22 (iii)
L.H.S.= 34.2p(l - g)(vim 2 - m m ) + 2(p + rq)(m3v2 - m27;3)
R.H.S.=------------- ---------------- ^ ~ ~
= 1 ^ = 34 216
Result is verified.
89
Following theorem gives- an expression for J„.
Theorem 23. Jn = - r J {QP)Hn_2 - (Pj [p> - J ^ )H n^ + (l - J ^ ) H n + J f ) .
Pvoof. Let Hn be homogeneous relation and be a particular solution of
equation (4.1). Then we have
Jn = Hn + = Clan + c20n + c37 n + J {np\ (4.17)
with the seed values, J0 = 0, J\ = 1 and J2 = p. Using these values and
solving we get,
Cl = 1 - (1 - j [ F)W + 7) + (p - j 'P ) \
= - a - - c) + o> - 4 P))]-A a
C2 = - (1 - ■ 'PX p - « + (P - j f ’)]
and
C3 = - (1 - A F))(v - 1 ) + (P - ’)]■A 7
Substituting C\ , c2 and C3 in equation(4.2) and with some computations,
we get
Hn = ^ - r 4 P)[an~1('y - 0) - £n_1(7 - a) + 7{n_1))(/? “ °01
-p(l - J,(P))[an(7 - 0 ) ~ 0n{ l - a) + 7(n)X£ " «)]
+(1 - J,(P))[an+1(7 - 0) - /?n+1(7 -<*)+ 7 (n+1))(/3 - a)]
+(p - ; r ))[a"(7 - /?) - 0n(l - «) + 7(n))(/5 " «)]
- p (l - 4 P))«n -i + (2 - + (P - J P W ^ -
Hence, solution of (4.1) is given by
Jn = - r J {0P)Hn- 2 - (p4 P) ~ A P))Hn- 1 + (1 - + A P)-
90
□
Similarly we can obtain expressions for the n*'1 term of the other two se
quences as
and
4 = r ( l - 4 p>) H ^ 2 + ( p j p + g - + Jf>
4 = ~ r 4 P>H l 2 + (p 4 P> + r - - 4 P>J% + JV).
Remark:
Similarly we can obtain expressions for the nth term of the other two sequences
as
4 = r(l - 4 P>) K -2 + (p 4 (p) + 9 - 4 P)) H l- l - 4 P)H l + 4 P>
and
4 = - r 4 P)H ^ 2 + (p4 P> + r - 4 P)))H l-i ~ J P # ! + 4 P)-
4.3 U se o f E -O perator
In this section we show that the pseudo Tribonacci relation can be reduced to
pseudo Fibonacci relation by means of E operator. This will help us to write
down number of identities for pseudo Tribonacci by using those of pseudo
EJn — Jn+!•
Fibonacci.
Define E-operator such that
91
Let o, 0, 7 be the distinct roots of auxiliary equation
x3 — px2 — qx — r = 0.
Then,
(x - a)(x - P)(x - 7 ) = (x2 - p x + q)(x - 7 ) =
where a + 0 — p and a0 — q.
Hence, from (4.1) the recurrence relation
Jn+3 — pJr1+2 d~ qJn+1 d~ f Jn d" At
reduces to
(E2 - p E + q ) (E - 'r ) J n = Atn.
Therefore,
(E2 — pE d- q)un = Atn
where (E — 7 )Jn — un-
Hence,
1ln +2 — P u n + 1 d" qun — Atn,
or
Un+2 = PUn+1 d" Qun + At ,n > 0,
with uo = 0 and u\ — 1 .
Various properties of un can be utilised for Jn.
(4.18)
92
4.4 G en era liza tion o f { Jn}
In this section we use Elmore s techniques mentioned in section(3.7) to extend
the pseudo Tribonacci sequence Jn to { Then using generalized circular
functions, a new sequence is obtained and is denoted by {K n(x)}.
Let
K0(x) = CiNr<0(a*x) + c2Nrfj{fi*x) + c3Nr> 0(7 *^) + ANr<0(t*x) (4.19)
where a* = a l r , = /31//r, 7 * = 7 1/r and t* = t1//r, r being a positive
integer, cj, C2, C3, and A are constants.
Now define the sequence {Kn(x)} successively as follows:
Let K i(x) = K q \ x ), K 2(x ) = K^2r\ : r),..., and in general
Kn(x) = Konr\ x ) , where derivatives are with respect to x.00 f(nr+j)
Since, NrJ = E 7 — - 7x7, j = 0, 1 , ...,r - 1 ; r > l„=o (nr + j)\
and N % \t) = Nrfi(t), n = 1 , 2,...
using equation (4.20), we get
K\(x) = ciaNrfl(a*x) + c2fiNTt o(fi*x) + 037^ 0(7 *) + A tN rfi(t*x)
K2(x) = c\a2Nrfi{o*x) + c2p2Nr,0(l3*x) + c ^ N ^ * ) + A t2Nrfi(t*x)
and in general
Kn{x) = cianNr<0(a*x) + c2/3nNr,0{/3*x) + c tfnNrfi(Y ) + AtnNrfi(t*x).
We have the following result.
Theorem 24. K n{x) satisfies the non-homogeneous recurrence relation
K n+3(x) = pKn+2(x) + gKn+1{x) + rK n{x) + At"
93
Proof. We have
^ ■ S .= p {c 1a n+2N rt0(a *x)+c2^n+2N rfi( ^ x ) + c ^ n+2Nrt0{1 *)+Atn+2Nrfi{t*x))
+ q(c1an+1Nr<0(a*x) + c2/3n+1 Nrfi(/3*x) + c3ln+1Nr< 0(7*) + Atn+1Nrfi(t*x))
+ r(c1anNrfi(a*x) + c20nNr<o(/3*x) + c37niVr> 0(7 *) + AtnNrfi{t*x)) + Atn
=Cian Nr<o(a* x )\pa2+ q a + r]+ c 2fin N r<0(/3* x)\p^2+q/3+r]+C3'yn Nrfilp'y2 Pq^+r]
+ tnNrfi\pt2 + qt + r + A]
Using the fact that, a , /?, 7 are the roots of x3 — px2 — qx — r = 0, we get,
R.H.S.=Cian+3Arr,o(a*ar) + c2f3n+3Nrfi(/3*x) + c37n+3Arr>o(7*) 4- Atn+3Nrfl(t*x)
= /fn+3(a:)=L.H.S. □
Remark:
Observe that if r = 1 , then a* = a , 0* — /?,7* = 7 , and hence Nr>0(x) = ex.
Hence for r = 1 , we have
Kn{x) = cie“x + c2e&x + c3e~>x + Aetx.
=J*(x), which is similar to Elmore’s generalisation as stated in section (3.7).
Further with p = 1, q = 1, r = 1 A, = 0 and x = 0, K n(x) reduces to Tribonacci
sequence.
94
Chapter 5
Pseudo Fibonacci Polynomials
5.1 In trod u ction
The study of Fibonacci polynomials is an important topic and has applica
tions in different fields. Fibonacci polynomials have been studied on a more
advanced level by many mathematicians [29],[30]. Majorie Bicknell [11] has
given plethora of identities for Fibonacci polynomials and studied their divisi
bility properties. Many of the identities for Fibonacci polynomials are straight
extensions of the similar identities for Fibonacci numbers [25]. As reported
earlier in section 3.4. Fibonacci polynomials Fn(x) are defined by
F0(x) = 0, F i(x) = 1, and for n > 2, Fn+2(x) = xFn+\(x) + Fn(x).
The first few Fibonacci polynomials are
F0(x) = 0, F\(x) = 1, F2{x) = x, F 3 {x ) = x 2 + 1, F 4 {x ) = x3 + 2x and
F5(x) = x4 + 3x2 -l- 1 .
Note that Fn(l) = F„, the nth Fibonacci number. A closed form expression
for the nth Fibonacci polynomial Fn(x) is
a ( x r - m rFn^ a(x) - P{x)
(5.1)
95
where
fy(r \ — X + V (x2 + 4) n( \ X ~ ^/(x2 + 4) , .------------2-------- , 0(x) = ----------------- L (5.2)
are the solutions of
p(A; x) = A2 - xA - 1 = 0. (5.3)
For negative indices, we have F_n(x) = ( - l ) n_1Fn(a;).
The pseudo Fibonacci sequence {#„} have been discussed earlier in Chapter
3. The combination of concept of pseudo Fibonacci sequence and Fibonacci
polynomial give rise to a new class of polynomials. We call them pseudo
Fibonacci polynomials. In this Chapter we attempt to develop theory of pseudo
Fibonacci polynomials. Some identities of these polynomials are stated and
proved.
Definition: We define pseudo Fibonacci polynomial by the recurrence relation
given by
gn+2(x,t) = xgn+i(x,t)+ gn(x ,t)+ A tn, n > 0 , A ^ 0 and t^ O ,a ,0 . (5.4)
with go{x,t) = 0 and gi{x,t) = 1 .
It is easy to verify the relation
9-n (x ,t) = -xg -n+ l(x ,t) + g-n+2(x,t) — At (5.5)
which defines pseudo Fibonacci polynomials with negative indices.
We can express each pseudo Fibonacci polynomial in terms of Fibonacci poly
nomials. First few pseudo Fibonacci polynomials are given below:
g2(x, t)=x + A,
96
gz{x,t)=x2 + 1 +,4(a; + *),
g4{x, t)=x3 + 2x + A( 1 + x2) + A t(x + 1),
g5(x, t)=x4 + 3x2 + 1 + A x (2 + x 2) + At( 1 + x2) + A t2(x + t), etc..
These polynomials in terms of Fibonacci polynomials can be written as
gi(x, t)=Fi(x),
g2(x,t)=F2(x) + AFi(x),
93{x ,t)=F3(x) + A(F\(x)t + F2(x)),
g4(x,t)=F4(x) + A(Fi(x)t2 + F2(x)t + F3(x)),
gs(x,t)=F5(x) + >l(Fj(a:)t3 + F2{x)t2 + F3(x)t + F4(x)).
Hence the nth term of pseudo Fibonacci polynomial can be written as
gn(x, t) = Fn(x) + A J 2 Fi(x)tn-(i+1'>. (5.6)i=1
At x = 1 , the pn(l.<) = 9n, the n th pseudo Fibonacci number. Further like
pseudo Fibonacci number, pseudo Fibonacci polynomials also consist of two
parts. First part consists of Fibonacci polynomial and the second part consists
of polynomial in t whose coefficients are A times Fibonacci polynomials. In
fact nth Pseudo Fibonacci polynomial is a polynomial of degree n - 1 in x and
a polynomial of degree n — 2 in parameter t.
The following tables give coefficients of xn, n > 0 in the first part and
the coefficients of A tn,n > 0 in the second part of gn(x,t) respectively. In
Table No. 1 , observe that the binomial coefficients are appearing diagonally.
In Table No. 2 entries are the Fibonacci polynomials. Both the observations
are as expected from relation (5.6).
97
n x° x1 x2 a;3 x4 X5 X6 X7
0 0
1 1
2 1
3 1 1
4 2 1
5 1 3 1
6 3 4 1
7 1 6 5 1
8 4 10 6 1
Table No.l : Coefficients of xn in first part of gn(x ,t)
n >4 A t At2 At3 AtA At5 At6
0
1
2 1
3 X 1
4 x2 + 1 X 1
5 x3 + 2x x2 + l X 1
6 x4 + 3x2 + 1 x3 + 2x x2 + l X 1
7 x5 + 4x3 + 3x a;4 + 3x2 + 1 x2 + 2x x2 + l X 1
8 :r6 + 5x4 + 6x2 + 1 x5 + 4x3 + 3x x4 + 3x2 + 1 x2 + 2x X2 + 1 X 1
98
Table No.2 : Coefficients of A tn in second part of gn(x,t)
5.2 S om e F undam ental Identities o f gn(x, t)
In this section we discuss some of the identities of pseudo Fibonacci polyno
mials. Following identities are in order.
(i) Binet type formula
For n > 0, we have
g„(x,t) = cian(x) + Oiff'Ui:) + zV \ (5.7)
where
2 — A2 - Ax - 1
We have
q(x) =x + v x 2 + 4 x — \/x 2 + 4
and p(x) — -------^ '
(5.8)
(5.9)
Also_ l - z ( / 3 - t) d = (t - a) z l (5 .10)
C l~ a - P a ~ POO
(ii) Generating function for gn(x ,t) is given by G*{s) = ^ g n { x ,t) s n
. v a 9.1
, provided |ts| < 1 . (5.11)(1 - xs - s2)
(1 — ts)s -f As1 1 — ts
99
Proof.
Let G*(s) — y gn(x ,t)snn=0 oo
= H l[9n+2{x, t) - xgn+1(x, t) - AH snn=0 J_ ~— 2s 9n+2(x, t)sn — X y gn+l(x, t)sn — A y tnSn
n==Q n=0 n=0— ,.-2 oo oo
S “ 9is l ~ x s r l l l 9 n ( x , t ) s n - A y ( t s ) n.” n—0n=0
Hence
G*(s)(l - xs - s2) = s + A s2 § (ts)nn—0
i r As2G'{s) =(1 — xs — s2) s +
1 — ts provided |ts| < 1 .
Therefore
G-(s) =(1 — xs — s2)
(1 — ts)s + As21 — ts provided |ts| < 1 .
□
Note:- For simplicity, we write (5.7) as gn(x ,t) = cjq” + c2(3n + ztn in
what follows.
(hi) limn-4oo9n{x,t)
9n-l(x, t)= a , if \t/a\ < 1 .
Proof. Using Binet formula (5.7), we have
ci a" -f- e2/?n 4- x tn c ian_1 + C2/5n_1 + x tn~x Ci a + c2(0 /a )n~1l3 + x{t/a)n~H
ti Sd a + c2 {j3/a)n~l + x{t/a)n~l
= a, provided \t/a\ < 1 .
,. 9 n (x ,t)lim -------—-rn->°°0n_i(:r,t)
100
□
Here we use the fact that
lim (—)n —> 0 as n —► oo and lim (—)nn—voo q 7i—►oo ot 0 as n -> oo.
(iv) lim' ' n_9n{x,t)S " "* ’ if l‘/“l < L
Proof. Similar to that of identity (iii) and hence omitted.
(v) £ 9k{x,t) = - {k=0 X
Proof.
gn+2(x,t) + (1 - x)gn+1(x,t) - 1 -
9n+2(x,t') -f- (1 x^gn+i(x,t') 1 — A{n + 1),
[ gn+2{x,t) + (1 - x)gn+i(x,t) - 1 - A( *=££■),
Y19k(x , <) = 5Z[5fc+2(a:, t) - xgk+1(x, t ) - ^ n]Jfc=0 fc= 0
n
Therefore
a;
Hence
E 9k(x,t) k=0
= Y l 9k+2{x,t) - x Y , 9k+i(x,t) - A j 2 t nfc=0 fc=0 &=0
n n9k(x , t) = 0„+i + fi-n+2 ~ 1 - ^ + 1 ~ A Y f
fc=0 *=0
<7n+2(*, <) + ( ! - ar)ffn+1(ar, 0 - 1 -
9n+2^x, t') ~h (1 x)gn+i (x, P) 1 A(n +1),
9n+2{x,t) + (1 - x)gn+l(x ,t) - 1 - A (*=££)t
We have the following version of Catalan identity. [25]
□
if |*| > 1 ,
if |i| = 1 ,
if |*| < 1 .
if |*| > 1 ,
if )*| = 1 ,
if |*j < 1 .□
101
Proposition 25. For n > 1 ,
9n+i(x,t)gn- i ( x , t ) - g l ( x ,t)
= ( - 1)" {1 + z(x2 + 4) - P(t;x)z2}+ ztn{tgn_1(x ,t)-2 g n(x ,t)+ t-1gn+1(x]t)},
where z = ■
Proo/. Using Binet form (5.7), we have
0n+i(*, t)gn-i(x , t) - gl(x, t)
=(ciQn+1 + c20n+1 + ztn+1){cla " - 1 + c2/3n-1 + z t"-1) - (cian + c2/?n + ztnf
=ClC2an -1/5n+1+ClC2Qn+1^n_1+2tn+1(cla"_1^-c2 l_'1)^-^tn_1(clan+1^-c2 n',"1)—
2c\c2anfin — 2ztn(c1an + c2/?n)
=c1C2an0n[(0/a - 1) + (a /0 - 1)] + z f+ ^C ia " - 1 + c2/?"-1) + z<w- 1(ciaB+1 +
C2 *+1) - 2z«n(c1a n + c2/?n)
=c,c2(ar/?)n-1(a - /3)2 + zin+15„_i + - 2in^n
= —(a/?)n{z2p(J; x) — z(x2 + 1 ) — 1 } + ztn+1gn- i + ^ n"*1fl,n+x — 2t”<?n
= ( - l ) n {1 + z(x2 + 4) -p { t;x ) z 2} + ztn{tgn- 1{x ,t)-2 g n(x ,t)+ t-1gn+1{x-,t)}.
□
102
The following version of Catalan identity generalizes the above result [25],
Proposition 26. For two integers n ,r with n > r, r > 1 ,
9n(Xi^) ~ 9n+ r{x , t )gn_r ( x , t )
= ( - l ) n_r F?(x){l + (x2 + 4)z - p(t-,x)z2} - z tn{trgn-r(x,t) - 2gn(x,t) +
p9n+r{x >t)}, where FT(x) is the rth Fibonacci polynomial and z = t2 v—7.
Proof. We use Binet formula (5.7) to prove this proposition.
Now gn(x,t} gn+r(x,t')9n-rix iP)
=(cian -f c2pn + z tn)2 - (a a n+r + c2pn+r + ztn+r){c1an~r + c2pn~r + ztn~r)
=2cjc2ttn/in + 2ztn(d a n + c2(3n) - [.ztn~r(Clan + c2pn) + ztn+r(Clan + c2pn) +
c1c2an+rpn~r + clC2an- rPn+r]
=c,c2anPn[2 - ar/ fir - /3T/a r] + 2ztngn - ztn~rgn+r - ztn+rgn
=c1c2(a/3)n -r[(c*r - Pr)2] + ztn[2gn - gn+rt~r - gn- rtr]
= ( - l )n- r Qr — 0ra — ft
(a — P ) 2 C\C2 + ztn[2gn — g n+ r t r 9 n - r t r]
= ( - l) " - r [Fr{ x ) f {1 + (x2 + 4)z - p(t; x)z2} - z tn{trgn- r(x,t) - 2g„(x,t) +
±gn+r(x,t)}. ar
The above two identities are generalized by the following identity which is
like Vajda’s identity[25].
Proposition 27. For n, i , j > 1,
gn+i{x, t)gn+j(x, t) - gn{x , t)gn+i+j(x> 0
_ (~ l)n Fi(x)Fj(x){ 1 + (a;2 + 4) z -p { t \x ) z 2} + ztn{tj gn+i{x,t) + tlgn+j(x,t) -
ti+ 9n{x > P) 9n+t+j{Xi^)}i
where F<(x) is the ith Fibonacci polynomial and z = A2_Ax_r
103
Proof. Using Binet formula (5.7), we write
9n+i{x>t)9n+j{x,t) — 9 n { x , t ) g n+i+j
=(cia +t+c20n+,+ ztn+')(cian+i-\-c20n+3+ztn+i) - (c 1an+c2l3n-hztn)(clan+i+j+
C2f t +i+j + Z t n+i+J )
= < ?ia2 n + i+ j+ C 1C2 a n + i0 n + j + C 1C2 a n ^ j3 ” + i + (% p 2 n + i+ j + z t n + i ( C i a n+J + C2p n +j ) +
ztn+j{cxan+i + c2/3n+i) + z2t2n+i+j
- [C jQ 2n+<+-' + C%02n+ i+ J + 2 ^ 2 n + t + j + C l c 2 Q n ^ n + i+ j + c ^ m + j ^ p n
+ztn(cxan+i+j + c2/3n+i+j) + z tn+i+i{cxan + o ftn)\
—cxo2(aP)n(aif t + f t f t ) + ztn+ign+j(x, t) + ztn+ign+i(x, t)
-\cxc2anfin+i+* + cxc2an+i+i f t 4- ztngn+i+j{x,t) + ztn+i+* gn(x,t)\
=cxc2(ap)n{(jt( a P -P ) - a l(aj -/3j )]+ztn[tign+j(x, t)+tj gn+i(x, t)-g n+i+j(x, t ) -
tii+j)gn(x , <)]
=c1c2(a^)n[(aJ - ft)(a* - f t ) + zfn[ffirn+i(x,t) + tj gn+i(x,t) - 9n+i+jOM) -
t(<+j)0»OM)]
= ( - l ) n Fi(x)Fj(x){l + (x2 + 4)z -p (£ ;x )z2} + z tn{V gn+i(x,t) + tign+j(x,t) -
t 9 n { x , t ) g n + i + j ( x , t)},
where .Fj(x) is the ith Fibonacci polynomial and z = A2_Ax_i - ^
Proposition 28. For n,m integers, we have
5m+i(x, t)gn(x , t) — gm(x, t)gn+i (x> t)
— i<’n_m(x){l + (x2 + 4)z — p(t',x)z2} + ztn{tgm(x, t) — <?m-l(x, £)}
+ztm{gn+x(x, t) — tgn(x,t)}.
where Fr(x) is the rth Fibonacci polynomial and z = A2_Ax- i '
104
Proof. Using Binet form (5.7), we get
9m +i{x,t)g„(x,t) - gm{x,t)gn+1{x,t)
=(c,Qm+1 + C2p + *») _ + C20m + 2(™)(cia„ „ +
C2/3"+l 4. ;r*»+l)
=c2am+„+l + ^m+n+1 + Z2t2n + +
+ztm+l{a a n + c2/3n) + 2£n(Clam+1 + c2pm+1)
-[cfcm+n+1 + c2/?m+n+1 + 22£2n + c1c2a mj5n+1 4- cic2a n+1/?m + «£m(cian+1 -f
c2 n+*) + ztn+1(ci a m + c2fim)
=c\C2acm[jn(a —(-})—cic2an(im(a —fi)-\-ztrn+lgn(x, t)+ ztngmjrl(x ,t)—ztmgn+1(x,t)—
ztn+1gm(x,t)
=CiC2 ( a 0 ) m ~ 1[ ( a n ~ m - (r 1"" ) + ztm+lgn(x, t) + ztngm+1(x, t) - n + i(® , t) -
ztn+l9m(x,t)
= ( _ l ) m- i Fn_m(x ) { i 4 ( x 2 + 4)^ -p (£ ;x )z 2} + 2 £”{£firm(a;,£) - 3 m_i(a;,£)}
+ 2 tm{gn+1(a:,£) - £i?n(x,£)},
where Fr (x) is the rth Fibonacci polynomial and 2 = □
5.3 P seu d o F ib on acci P olynom ial in two vari
ab les
. In this section we extend the pseudo Fibonacci polynomials to obtain poly
nomials in two variables.
Definition: We define pseudo Fibonacci polynomial for two variables by the
105
recurrence relation
(5.12)with go(x,y;t) = 0 and gi(x,y,t) = 1.
are the roots of
p(X] x, y) = A2 - xX - y = 0. (5.13)
Here, a + = x, a - /3 = f i x 2 + 4xy), afi = -y .
The sequence can be extended to negative integers by defining
9-n(x,y,t) = -Z g _ n+l(x,y;t) + ~g .n+2(x,y;t) - f t~ n.
First few terms of gn(x,y) are
9o(x, y,t) = 0,
9i{x,V\t) = 1,
92(x,y;t) = x + A,
g3(x, y;t) = x 2 + y + Ax + At,
9*{x, y, t) = x 3 + 2xy + (x2 + y)A 4- A xt + At2,
g$(x, y; t) = x 4 + 3x 2y + y2 -f (a:3 + 2xy)A 4- (x2 4- y)A t 4- A xt2 + At3.
Observe that like pseudo Fibonacci polynomials in single variable, these poly
nomials also have two parts.
We now look at some fundamental identities for pseudo Fibonacci polyno
mials in two variables.
(i) Binet formula for gn(x ,y , t ) is given by
9n(x,y,t) =\(x + y ) - A t ] - ( l - z ) P nn (
a — P{x + y ) - A t ) pn + zfn
106
where
A2 - V - X x - f (5.14)
(ii) Generating Function for gn(x, y; t ) is given by G*(x, y; t) = § yn(^) t)sn,oo w=0
S + As2 51 (fs)n_ ______ n=0____1 - xs — ys2 ’
5(1 - ts) + As2(l - (ts)n+1)- (1 — ts )(l — is — ys2) '• provided l*sl < L
('“> J .9 « ( i ,! / i( ) = ■ ■_ * _ fl - yg„(x,y,t) - 9„+1(i,!/;() + .4 £ tk .fc-o i x y l fe=0
Proof. From the recurrence relation (5.14), we write
E 9k+2(x, y \t) = x £ yfc+i (x, y;f) 4-y E yfc(z, y;0 + ^ E £fc-*=0 *=0 fc=0 fc=0
Thereforen+2 n+1 n nE Sfc(z, y; 0 = * E &(*> y; 0 + y E 5*0*:, y; 0 + -A E £*•fc=0 fc=0 fc=0 Jfc=0
i.e.n nE 9k(x,y\ 0(1 - x - y ) = 9 i - 9n+\ - 9n+2 - xgn+i{x,y\t) + A £ t*.
Jk=0 fc=0
HencenE 9k{x,y\t)
k=01
(1-i-y) l y9n+\{x■) yt 9n+i(.x,y,t) A E £fc=0
□
(iv) E 9k (x ,y ;t) tkk=0
£ - gn+i(x, y; £)£n+1(*£ - 1) + fln+afr, y; 0*n+2 4 £ ^ + 21 — £x — y£2 fc=o
Prw/. L .H.s.= E ( x y Jk- i ( ^ y ; 0 + yfe-2(^y;i ) + ^ fc 2) f 'k=0
= E {xgk-i{x , y; £)£* + gk-2(x, y; £)£fc + A tk~Hk)fc=0
=x E yfc- 1 (x, y; £)(x, y; t)tk + fc5 0 9k—2[X, y; t)tk + A j : *
=x E 9k (x ,y ,t) tk+1 + E 9k(x,y-,t)tk+ ^ feE *
2k—2
k= - 1
i2fc—2
107
- * r « ) + Eo^ ( x ,y ; t ) t ‘+> + + g^ yit)t _
9 n-i(x, V, t)t — gn(x, y; t)tn+2 + £} (gk(x, t)tk+2 -{•AY' t2k~2k=° feto
gn-i(x,y,t)tn+1 - gn(x,y,t)tn+2 + t2 £ gk{x,y,t)tk + A £ t2k~2.fc=0 fc= 0
Hence,
(1 - - t2) ^ g k(x ,y , t) tk = xg_l (x,y,t) + g_2(x,y;t) + At~2 - A r 2 -
(Z0«(z,y;*) + 0n-i)Or,2/; t)fn+1 ?/; i)i — gn(x, y;t)tn+2 + A £ t2k~2.k=0
=0o(x, v \ t ) - A t 2 - (xgn(x, y; t) + n_x(a;, t/; £) + ^ n-1)in+1 + +
9-i{x,y,t)t - gn(x ,y ;t)tn+2 + A £ t2*-2.k=0
Therefore L.H.S.
= ( i 1 'te - t2) ^~A t~2+At2n~ 9n+\{x, y \t)tn+l+{gl {x, y; t)-a£ _1)-£rn(a;, y; t)tn+2+
A t t2k~2.k=0
_ t - 9n+i(x,y;t)tn+1(xt - 1) + gn+2{x,y;t)tn+2 1 — tx — yt2
This completes the proof.
- A E t2k+2.k=0
□
Following result is Catalan’s type identity in two variables.
Proposition 29. For n,m integers, we have
9n+r(x, y, t)gn-r(x , y; t)—g2(x, y; t) = ^= & jL(-y)n- r{z ( 2 t - x ) - l - z2p(t\ x, y)}+
ztn{gn+r(x, y; t ) / t r 4- gn- r(x, y, t) tr - 2 g„(x,y;t)}.
Proof of this result is similar to the proof of Proposition 26 and hence
omitted. Following result follows immediately with r = 1 in above result.
108
Proposition 30. For n,m integers, we have
Sn+i(*, V\ t)gn- 1 (x, y, t) - gl(x, y; t) = { - y )71' 1 {z{2t - z ) ~ 1 - z2p(t, x, y; t)} +
ztn{gn+\/t + gn-\t - 2gn{x,y]t)}.
109
Chapter 6
Congruence Properties o f Gn
6.1 In trod u ction
In this chapter a generalized pseudo Fibonacci sequence with particular value
for the parameter t is considered with an aim of possible application in areas
such as Financial analysis. Some well known identities for this sequence are
obtained by using identities of Generalized Fibonacci sequence. We also study
some of the congruence properties of this sequence. Congruence properties of
Fibonacci sequence can be seen in [27] ,[21].
Let p,q € Z and A be constant such that 1+ p — q ^ 0 andA
u> = ----------- e Z. Consider the Pseudo Fibonacci Sequence defined by the1 + V - <1
recurrence relation
Gn+2= pG n+1 + qG n + A { - l ) n (6.1)
with Go = uj, G i = 1 — w.
Equation (6.1) generates generalized pseudo Fibonacci numbers with param
eter t = -1 . Some well known identities for G„ are obtained by using the
identities of generalized Fibonacci sequence {Pn} defined by
Pn+2 = pPn+l +Q Pn (6'2)
110
with Po = 0 , Pi — 1 .
Following result follows immediately.
Theorem 6.1.1. The nth term Gn of (6.1) is given by Gn = Pn + w (- l)n
where P„ satisfies (6.2).
Proof. Let Gn — Pn + B{ — l)n be a solution of (6.1). Then on substituting in
(6.1), we get B = ^ = u>. Hence the result. □
Following identities of (6.2) are found in [1].
(a). I Pr =r=0qPn+Pn+i -1
p+q-1
(b). P„2 - P „ +1P„_l = ( -g ) " - 1
(c) . Pn+m-1 — gPn—\Pm— 1 "b PnPm-
(d) . C ata lan ty p e identity : P 2 - Pn+rP„_r = (-<?)n-rPr2.
(e) . d ’O cagne ty p e identity : P„+iPm - PnPm+1 = (-q )nPm-n-
111
6.2 Som e Id en tities
In this section we obtain some important identities of the sequence {Gn}. We
have the following.
Theorem 31. Gn satisfies following identities.
A- _ qGn + Gn+l - u ( - 1) ( g - l ) - lV l*r=ou r ----------------------;-------r—--------------1- 0J€n where
P + 9 ~ 1
0, i f n is odd,
l , i / n is even.
ii) Gn+iGn-.i — G* = (—1)" qn~l ~ w(—l)n(Gn-i + 2 Gn + Gn+i)
Hi) Gn+1Gm - GnGm+l = (-q )nGm-n + w[(G„+1 + (?„ )(-l)m +
(Gm+i - Gm) ( - l ) n - ( ( - 9)n + 2(—l)m+n)]-
tt^ G ^-G u + rG n -r = (—9)n -rG^+cu[2Gn —(—1) rGn+r ~~(~ 1) G„_r](—1) +
(~q)n~ru>2 - 2 w (-9 )n- r ( - l ) r Gr .
Proof, (i)
£ c r = E P , + w ( - i ) 'r = 0 r = 0
= E P r + « E ( - l ) 'r = 0 r = 0
= gPn + ^n+l.I J 1+aJenp + 9 - 1
g ( G n - U ) ( - 1 ) W) + G n + l - -------- 1 1 1 4- CJ£i
— p + 9 - 1_ qGn + Gn+i — 1) (.Q~ A)---- I— p + 9 _ 1
112
where
00
/
—0, i f n is odd,
1 , i / n is even.
LH S - (P„+i + w (- l)n+1)(Pn_1 + v ( - l ) n- 1) - (Pn -h cv(-l)n)
— (^n+ lPn-1 ~ Pn) ~ U>( — l)n(Pn+i + Pn_j 2Pn)
— ( — 1 ) (—9) — u>(—l)n(Gn-fi + Gn-i + 2Gn)
= P //5 .
(iii)
LHS = (Pn+1 + u ;( - l)n+,)(Pm + cu(-l)m) - (Pn + u ( - l ) n)(Pm+1 + W(-l)™+1)
= (Pn+lPm - P„Pm+i) + u;(P„+1( - l ) m - Pro(-1 )B) + w (P „ (-l)m - Pm+( - l ) n)
= (~q)nPm-n + w(Pn+1 + P n )( - l)m + w(Pm+1 - P n ) ( - l ) n
= (-<7)"Gm_„ + u(Gm+\ - Gm)(—l)n + u(Gn+1 + Gn) ( - l ) m ~ u[(-q)n + 2(—l)m+n)]
= RHS.
(iv)LHS = [Pn + a ; ( - l )n]2 - [Pn+r + u j(-l)n+r][Pn- r + w (- l)”- r]
= Pn2 - Pn+rP„_r + 2PnU>( —1)" - Pn+ M - 1)”'" - Pn-rW (-l)n+r
= (-9 )n“r[Grr - w ( - l ) r]2+ 2w( - 1)n[C!rl- w ( - l ) n] - w ( - l ) n" r[Gr„+ r-w (-l)n+r] -
a ;(-l)"+r[Gn+r - ^ ( - l ) n+r]
= (~q)n~rGj + u[2Gn - { - l ) - TGn+r - ( - l ) rG „-r](-l)n +
{-q)n~ru 2 - 2u/(-<7)n- r( - l ) rGr .□
= RHS.
113
6.3 M odular P rop erties
Theorem 32. Let tt(m) be the period of Gn modulo m. Let e > 1 be qi
Thengiven.
i) For odd prime p, „ („ « ) = p ~ V ( p ) , where 1 < e < e is maximal so that
n(jf ) = tt(p ).
ii) For p = 2 and e > 2,/
7r(2*) = 2e e 7r(4 ), where 2 < e < e is maximal so that 7r(2e ) = tt(4).
Proof. Let 7r (rn) be the period of {Pn} modulo m. 7r’ (m) is always even.
Now G0 = Po + B, Gi = Pj - £?. Thus G0 = B and Gi = 1 - B.
Hence
G*'(m) = ^r'(m) + (m) = B ( mod rn) and
G»'(m)+1 = (m)+1 = 1 - B( mod m)
so that the period tt' {m) of Pn and 7r(m) of Gn are same.
Now the theorem follows from [[21], Theorem 2]. □
Next we consider a particular case of equation (6.1 ) with p = 1 , q = — 2
and i4 = 1 . Congruence properties of these Gn’s for specific values of m are
calculated as follows. First we show that the period 7r (rn) of Pn is even for
m > 2. By [21] ordm(q)\n (m). In our case q = —2, so that ordm(2)|II (m).
Now clearly for m = 3,5,7,9,11,13,15... ordm(2) is even. So that 7r'(m) is
even. Hence period 7r(m) of Gn is also even.
Next, we show that i f , Gr, Gr+l, Gr+2 modulo m are same as G„ Gs+U Gs+2
modulo m respectively , then we must have Gr+k = Gs+A;(modm), Vfc _ 3
114
This can be seen as follows: First note that it suffices to show that
G > + 3 = Gr+2 - 2Gr+1 + ( - l ) r+ 1
= Ga+2 ~ 2Gs+i + (—l)r_fl
= g s+2 - 2Ga+1 + (-1 y +\
— Gs+3
Note that r and s has same parity as the period n(m) of Gn is even.
Remark: The above argument shows that if three consecutive values of
Gn modulo m are same, then the remaining values also repeat. This is in
contradiction to Fibonacci sequence where two consecutive values of Fn modulo
m are same then the remaining values repeat, in what follows we take Gn’s
defined by Gn+2 = G„+i - 2Gn + ( - l ) n and m = 3 ,5 , 7 , 9 .
The table below gives Gn modulo 3
n 0 1 2 3 4 5 6 7 08 9 10 11 12 13 14 15
Gn 0 1 2 -1 -4 -3 6 11 0 -23 -22 23 68 21 -144 -157
Gn modulo 3 0 1 2 2 2 0 0 2 0 1 2 2 2 0 0 2
From above table and the Remark we get the following result.
Proposition 33.
0
G„ = * i
2
mod 3 i f n = 0,5,6 mod 8,
mod 3 i f n = l mod 8,
mod 3 i f n = 2,3,4,7 mod 8.
115
The table below gives Gn modulo 5
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Gn 0 1 2 -1 -4 -3 6 11 0 -23 -22 23 68 21 -114 -157
Gn Mod 5 0 1 2 4 1 2 1 1 0 2 3 3 3 1 1 3
n 16 17 18 19 20 21 22 23 24 25
G„ 72 385 242 -529 -1012 45 2070 1979 -2160 -6119
Gn Mod 5 2 0 2 1 3 0 0 4 0 1
FYom the above table and the Remark we can conclude the following.
Proposition 34.
G(n)
0 mod 5 i f n = 0,8,17,21,22 mod 24,
1 mod 5 * / n s 1,4,6,7,13,14,19 mod 24,
< 2 mod 5 i f n = 2 ,5 ,9,16,18 mod 24,
3 mod 5 i f n s 10,11,12,15,20 mod 24,
4 mod 5 i f n s 3,23 mod 24.
116
The table below gives Gn modulo 7.
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Gn 0 1 2 -1 -4 -3 6 11 0 -23 -22 23 68 21 -114 -157
Gn Mod 7 0 1 2 6 3 4 6 4 0 5 6 2 5 0 5 4
n 16 17 18 19 20 21 22 23 24 25
Gn 72 385 242 -529 -1012 45 2070 1979 -2160 -6119
Gn Mod 7 2 0 4 3 3 3 5 5 3 6
From the above table and the Remark we can conclude the following.
Proposition 35.
G(n) =
0
1
2
< 3
4
5
6
mod 7, i f n = 0,8,17,39,40 mod 42,
mod 7 i f n=. 1,26,28,33,35,36 mod 42,
mod 7 i f n = 2,11,13,16,27,30,31 mod 42,
mod 7 i f n = 4,19,20,21,24, 29, mod 42,
mod 7 i f n = 5 ,7 ,15,18,34,38 mod 42.
mod 7 i f n = 9 ,12,14,22,23,37 mod 42.
mod 7 i / n = 3,6,10,25,32,41 mod 42.
117
Table showing Gn Modulo 9
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Gn 0 1 2 -1 -4 -3 6 11 0 -23 -22 23 68 21 -114 -157
Gn Mod 9 0 1 2 8 5 6 6 2 0 4 5 5 5 3 3 5
n 16 17 18 19 20 21 22 23 24 25
Gn 72 385 242 -529 -1012 45 2070 1979 -2160 -6119
Gn Mod 9 0 7 8 2 5 0 0 8 0 1
From the above table and the Remark we can conclude the following.
Proposition 36.
G(n)
0 mod 9 i f n = 0,8,16 mod 24,
1 mod 9 i f n = 1 mod 24,
2 mod 9 i f n = 2,7 ,19 mod 24,
3 mod 9 i f n = 13,14 mod 24,
' 4 mod 9 i f n = 9 mod 24,
5 mod 9 i / n = 4 , 10,11,12,15,20 mod 24,
6 mod 9 i f n = 5,6 mod 24,
7 mod 9 i f n = VI mod 24,
8 mod 9 i / n = 3,18,23 mod 24.
118
SUMMARY
This study deals with a new concept of pseudo Fibonacci sequence which is
a Fibonacci like sequence. We consider here a second order non-homogeneous
recurrence relation as an extension to homogeneous recurrence relation defin
ing the well known Fibonacci sequence. Some interesting identities related
with this {G„} denoted by {gn} are stated and proved. Later this sequence is
extended to a new sequence {Gn} by altering the coefficients of the terms in
the sequence. Some identities and theorem concerning the new sequence are
proved. Among these identities, we have Binet formulae, Cassini’s identities,
Catalan’s identities, d’ Ocagnes identities and summation of terms of the se
quences in different forms for {gn} and {Gn}. We illustrate the results obtained
for {£?„} by giving some examples. Further extension of {G„} is achieved by
introducing Elmore’s techniques and generalized circular functions. The well
known Binet formula, matrix method and other techniques are used to prove
different results. This is the content of chapter III.
In chapter IV, the sequence {Gn} is extended to get pseudo Tribonacci se
quence by considering the third order non-homogeneous recurrence relation.
Results concerning generating function, Binet formula and summation of n
terms of the said sequence are obtained. Some results are illustrated with ex
amples.
A new class of polynomials called pseudo Fibonacci polynomials are studied
in chapter V. We have studied different properties and proved some results for
119
single variable and bivariate cases of these polynomials.
Chapter VI deals with congruence properties of {Gn} with particular values for
the constants and the parameter. Some properties of Gn modulo odd number
are obtained.
Chapter I is Introduction and Chapter II gives extensive survey of exiting re
sults on Fibonacci sequence. This particular study was taken with a view that
application of {Gn} will find place in Financial analysis and other fields where
ever Fibonacci sequence is applicable.
Problems for further study
First problem that comes into picture is to extend these ideas to r*h order
recurrence relation to give pseudo r-bonacci sequence. One can also think of
applying matrix methods to obtain various other results. Congruence proper
ties modulo some even number are to be explored. Most important problem
is to find proper application of the concepts studied in this work.
120
List of Publications/ Communications Based on the Thesis.
PI Phadte C.N., Pethe S.P. “On second order non homogeneous Recurrence
Relation” Annales Mathematicae et Informatcae, Vol.41 (2013), pp. 205-
210.
P2 Phadte C.N. “Extended Pseudo Fibonacci Sequence” Bulletin of the
Marathwada Mathematical Society, Vol.15, No.2 (2014), pp. 54-57.
P3 Phadte C. N., Pethe S. P. “Trigonometric Pseudo Fibonacci Sequence”
Notes on Number Theory and Discrete Mathematics, Vol.21, No.3 (2015),
pp.70-76.
P4 Phadte C.N., Valaulikar Y.S. “Pseudo Fibonacci Polynomials and Some
Properties” Bulletin of the Marathwada Mathematical Society, Vol.16,
No.2 (2015), pp. 13-18.
P5 Phadte C.N., Valaulikar Y. S. “Generalization of Horadam’s Sequence”
Turkish Journal of Analysis and Number Theory, Vol.4, No.4 (2016), pp.
113-117.
P6 Phadte C.N., Valaulikar Y.S. “On Pseudo Tribonacci Sequence” Inter
national Journal of Mathematics Trends and Technology Vol.31, No.3
(2016), pp. 195-200.
P7 Phadte C.N., Tamba M., Valaulikar Y.S. “Congruence Properties of a
Pseudo Fibonacci Sequence” Communicated.
121
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