ELSEVIER Journal of Statistical Planning and

Inference 54 (1996) 101 118

journal of statistical planning and inference

A unified approach to the summation and integration formulas for q-hypergeometr ic functions I 1

M i z a n R a h m a n a,,, Serge i K. Sus lov b a Department of Mathematics and Statistics, Carleton University, 1125 Colonel By Drit:e, OHawa.

Ontario, Canada KIS 5B6 b Russian Research Centre Kurchatov Institute, Moscow 123182, Russia

Received 10 February 1994; revised 23 November 1994

Abstract

The most basic summation formula in the theory of q-hypergeometric functions is the well- known q-binomial formula. Not so well-known is the fact that there is a bilateral extension of it due to Ramanujan, and that there are two integral analogues of it. We show that these summation formulas as well as their integral counterparts have essentially the same origin, namely, a Pearson- type difference equation on a q-linear lattice. It is shown that the boundary conditions determine the structure of the solution of this equation which also enables us to evaluate the sums and integrals by a systematic process of iteration. We conclude by giving a very simple derivation of the q-Gauss formula and a second summation formula for a nonterminating a(/~ series.

A M S class!Jication: 33D05; 33D15

Keywords: Pearson equation; Summation and integration formulas; Basic hypergeometric series

1. Introduction

The hypergeometric equation is a linear second-order homogeneous differential equa- tion of the type

~ ( x ) y " ( x ) + ~(x)y ' (x) + 2y(x) = o, (1. I )

where a(x) and z(x) are polynomials of degrees at most 2 and 1, respectively, and are independent of the parameter 2. One important property of this equation is that if we

denote vo(x) := y(x), vk(x) := y(k)(x), the kth derivative of y, then vk(x) satisfies an equation of the same type, namely,

! c~(x)v~'(x) + rk(x)v~(x) + ~kv~(x) = 0, (I .:?)

* Corresponding author. J This work, supported in part by the NSERC grant # A6197, was completed while the second author was

visiting Carleton University in September~October 1993.

03'78-3758/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 3 7 5 8 ( 9 5 ) 0 0 1 60-3

102

with

M. Rahman, S.K. Suslov/Journal of Statistical Plannin9 and Inference 54 (1996) 101-118

rk(x) = z(x) + k~'(x), k ( k - 1) ,,

p~ = 2 + kz ~ + - a . (1.3) 2

When a ( x ) = x(1 - x ) , z ( x ) = c - (a + b + 1)x and 2 = - a b , a solution of (1.1) that is convergent for Ix[ < 1 is the well-known Gaussian hypergeometric function

O 0

2Fl(a,b; c ;x) := Z (a)n(b)n .:0 (c).n! x",

(1.4)

where the shifted factorials inside the sum are defined by

1 if n = O , ( a ) n = a ( a + 1 ) . . . ( a + n - 1) if n = 1,2,. . .

(1.5)

Most special functions of practical interest to the physicists, engineers and the statisti- cians are special or limiting cases of 2Fl(a, b; c;x) .

Formally, Eq. (1.1) can be written in a self-adjoint form

[p(x)f f(x)yt(x)] ' + ,~p(x)y(x) = O, (1.6)

where p(x) satisfies the Pearson-type equation (Pearson, 1895; Fisz, 1963)

(pa) ' = pz. (1.7)

Pearson pointed out that many probability density functions of statistical interest are special cases of solutions of this equation provided, of course, that p(x) is nonnegative on the interval of possible values of the corresponding random variable. Recently it was pointed out by the authors (1994a) that if a and/or z has at least one free parameter then this equation often enables us to normalize the distribution, in other words, to integrate over the function p(x). This is the key idea we shall employ in this paper to generalize the Pearson equation in a certain manner, to find the corresponding solution, and to determine the total weight by either summing an infinite series or by integrating a positive-valued function. From a statistical point of view the generalization that we are alluding to is motivated by the fact that the discrete distributions, like the binomial and Poisson distributions, do not arise from a first-order differential equation, rather from a first-order difference equation. So, we shall first consider a rather general sort of discretization of Eq. (1.1) studied extensively by the Russian school of Nikiforov and Uvarov, see, for example, Nikiforov et al. (1985):

8(x(s) ) [y ( s + h) - y ( s ) y ( s ) - y (s - h)]

x ( s + h / 2 ) - x ( s - h / 2 ) Lx(s+h) x(s) - x ( s ) ~ ; G J +~(x(s)) [y ( s + h ) - y ( s ) Y ( s ) - y ( s z h ) ] + 2 y ( x ) = O,

[x(~ + h) - - x ~ + x(~) x(~ - h) J (1.8)

M. Rahman, S.K. SuslovlJournal of Statistical Planning and Inference 54 (1996) I01 118 1(13

where x(s), assumed to be a continuously differentiable function of s on some domain of

the complex plane, defines a class of lattice with variable mesh-size Ax = x ( s + h ) - x ( s ) . The coefficients ~(x(s)) and g(x(s)) are, as in (1.1), polynomials of degrees at mosl

2 and 1, respectively, in x(s). It is clear that (1.8) is symmetric in h and that it approximates (1.1) to second order in h. By rescaling the variable s --, sh and using the notations

A x ( s ) = x ( s + 1 ) - x ( s ) ,

Vx(s) = x ( s ) - x ( s - 1 ) = Ax( s - 1 ),

x,~(s) = x(s +/~/2),

we can rewrite (1.8) in the form

1 VAy(s) a(x(s))v@l(s ) LAy(s)] + 5 "~(x(s)) L A x ( s ) ] L A x ( s )

Since

,lAy(s) Vy(s)] Ay(s) Iv[ay(s) l L a x ( s ) + V x ( s ) ] - A x i s ) 2 [ A x ( s ) J '

Eq. (1.10) can be written in a more suggestive form:

V@l(s) , , Ay(s) LAy(s)] + ts' + y(s) = 0, L Ax(s) J

where

• ( s ) = ~(x(s) ) ,

~ ( s ) = ~ ( x ( s ) ) - ½ ~(x(s ) ) V x l ( s ) .

If' p(s) satisfies the Pearson-type difference equation

A[p(s)a(s)] = p(s)r (s )Vxl (s ) ,

(1.,-))

that is,

V y ( s ) ] + 2y(s) = O. (1.10) + Vx(s ) J

1.11)

~.12)

1.13)

p(s + 1) ~r(s) + z ( s ) V x l ( s ) - 1 . ~ 4 )

p(s ) ~(s + 1) '

then (1.11) can be expressed in the self-adjoint form

v I , ,Ay(s)] p(s + 1)a(s + l ) ~ j +,L p(s)y(s) = O, 1.15) Vxl(s)

k

similar to (1.6). Our interest in this paper are solutions of (1.13), rather than of (1.15), for various

choices of the lattice x(s) and the coefficient functions a(s) and r(s). However, it is important to mention that a necessary and sufficient condition that (1.11) retains its hypergeometric property, namely, that the successive difference-derivatives Vk+l(s) :----

104 ~ Rahman, XK. Suslov/Journal of Statistical Planning and Inference 54 (1996) 101-118

Vvk(s)/Vxk(s), Vo(S) := y(s) also satisfy an equation similar to (1.11), is that the

lattice has the form

Caq -s + Czq s if q # 1, X ( S ) (1.16)

C~s 2 + C2s if q = 1,

where CI and C2 are arbitrary constants, not both zero. For a proof o f this characteri-

zation theorem see Atakishiyev et al. (1995).

The lattice variable s may be discrete, varying in unit steps from s = a to s = b - 1

(it is permissible for a to be -cxD and/or b to be +cx~, also a may be complex). In

this case (1.13) leads to a summation formula

b- -1

p(b)a(b) - p(a)a(a) = Z p(s)z(s)Vxl(s). (1.17) s~a

I f a and b are both finite then usually p(b)a(b) and p(a)a(a) are both zeros, in which

case one gets

b - - I

Z p(s)v(s)Vx~(s) = 0. (1.18) s=a

I f l i m a ~ _ ~ cr(a)p(a) = 0 and l i m b ~ a(b)p(b) = 0 then we have

o c

Z p(s)z(s)Vxl(s) : 0, (1.19) 8----0<3

provided that the bilateral series converges. One of the objectives o f this paper is to

show that this is the origin o f most o f the summation formulas for basic hypergeometric

series.

On the other hand, s may be a continuous variable in some domain o f the complex

plane in which case we may be able to find a contour C that does not pass through

the singularities, if any, o f either side o f (1.13). This leads to the formula

f c A[p(s)a(s)] dS = f c P(S)Z(s)Vx, (s) ds. (1.20)

I f C' is the contour obtained from C by the shift s ' = s - l, and if p(s)a(s) has no singularities between C and C', then, by Cauchy's theorem, the integral on the left

side o f (1.20) vanishes under proper conditions. This leads to the formula

cP(S)'C(s)VXl (s) ds = O. (1.21)

In particular, C may be the whole real line. This formula then becomes

f ~ p(s)~(s)Vxl (s) ds --- 0, (1.22)

M. Rahman, XK. SuslovlJournal of Statistical Planning and InJerence 54 (1996) 101 118 1(15

which may be considered as integral analogue of ( l .19), provided, of course, that

f ~: p(s )a(s ) ds < oc. (1.23)

In Rahman and Suslov (1994b) and Ismail and Rahman (1995) such integrals were

called Ramanujan-type.

The second type of integrals that has been considered in the literature corresponds to

C being the whole imaginary line or a line parallel to it. Watson (1910), Barnes (1908, 1910), Agarwal (1953) and others studied such integrals. However, since qi(S~2n' log q 't

: @', s ~ R and 0 < q < 1, integrals along the imaginary axis from s = - n / l o g q- ~ to

n / l o g q - t have also been considered by many authors when the integrands of (1.20)

depend on s through q+ only. See Slater (1952a, b, 1955, 1966) and Rahman and Suslov

(1994a, b). Since two of the most basic formulas for integrals along the imaginary axis or parts thereof were found by Barnes (1908, 1910), it was thought appropriate to call

them Barnes-type integrals.

In order to keep this paper to a reasonable length we have chosen to restrict ourselves

to the q-linear lattice x(s) = q±'~, and to the most basic summation and integration

fotTnulas that contain no more than three parameters. In Section 2 we give a derivatkm

of the familiar q-binomial formula based on a Pearson-type equation. In Section 3 we

obtain the Ramanujan 1~1 sum as well as its integral analogue. Section 4 is devoted to a Barnes-type integral for essentially the same p function. In Section 5 we employ

the same method to derive two summation formulas for the 2q51 function.

2. q-linear lattice x(s) = qS: the q-binomial formula

As far as the solution of the Pearson equation (1.13) is concerned it is clear from (1.14) that the important functions are a(s) and a ( s ) + r (s )Vx l ( s ) . Accordingly, let us

choose

a(s) : (1 - qS)qS, (2.[)

a(s) + z ( s ) V x t ( s ) : z(1 - aq")q s,

where a and z are some complex parameters. Note that a(0) = 0 and l i m s ~ a(s) : 0

when ]ql < 1. Eq. (1.14) then gives

p(s + 1) 1 - aq s p(s) -- 1 -qS-2~l (z/q). (2.2)

The solution o f (2.2), unique to within a multiplicative unit-periodic function (taken

to be 1 in the following), is

(a; q)+~ "z" ,s p ( s ) - - ~ ( / q ) , s ¢ - 1 , - - 2 . . . . . (2.3)

106 M. Rahman, S.K. Suslov/Journal of Statistical Plannin9 and Inference 54 (1996) 10l 118

where the q-shifted factorial is defined by

(a;q)s = (1 - a ) ( 1 - a q ) . . . ( 1 - a q s-l)

I f s is not a nonnegative integer then one defines

(a; q)o~ ( a ; q ) s - - s , s E C, (aq ; q)~

provided aq ~ ¢ q-k , k = O, 1 . . . . . and that

if s = 0, (2.4)

if s = 1,2 . . . .

(2.5)

(a; q ) ~ = lim (a; q)n (2.6) n - - - + o o

exists (which is ensured by the condition 0 < q < 1, that we shall assume to hold

throughout this paper). Our objective is to evaluate the sum

( x ) 0<3 (a; q)~ f ( a ) := Z P(s)qS = ~-" s (2.7)

s ~ 0 ~

It is obvious that the series is convergent for ]z[ < 1 and that the sum-function f (a) , considered as a function o f a, is analytic in any finite disc enclosing the point a = 0.

The infinite series in (2.7) is the special case r = 0 o f the basic hypergeometric series

r+J ~br defined by

[ a l , a2 . . . . . ar+l ;q,z] =~-~(a,,a2~__h_~_~ar+l;q)n (2.8) rq-l ~)r bl,bz . . . . ,br n:0 ( q ' - b ~ ' " " v " q ) n Z n '

where (al, a2 . . . . . ak; q)n = (al; q),(a2; q ) , . ' . (ak; q), . For the notations, definitions and

conditions o f convergence o f such series see Gasper and Rahman (1990).

Use o f (2.1) and (2.3) in (1.13) gives

A[p(s)a(s)] = [z(1 - aq s) - (1 - q)S]p(s)qS

_ 1-aa [1 1---a-z(1-aqS)]p(s)qS,l_a a • 0 , 1 . (2.9)

I f a = 1 then, since (1;q)~ is 1 when s = 0 and 0 when s = 1,2 . . . . . it follows from

(2.7) that f ( 1 ) = 1. Also, the restriction a ¢ 0 is not essential since it can be removed by simply multiplying both sides o f (2.9) by a.

It follows from (2.1) and (2.3) that p(0)a(0) = 0 and that l im~o~ p(s)a(s) = 0 if

Iz[ < 1. Since (1 - a q S ) ( a ; q)~ = (1 - a ) ( a q ; q)~, we then deduce from (2.3) and (2.9) the recurrence formula

f ( a ) = (1 - az)f(aq). (2.10)

Iterating it n - 1 times we get

f (a) = (az; q)nf (aq") (2.1 1 )

M. Rahman, S.K. SusIovIJournal of Statistical Planninq and Inference 54 (1996) lOl 118 107

which, in the limit n ---* oc, gives

f ( a ) = (az; q)~f (O) . (2.12)

This implies that

f ( a ) f ( 0 ) - - - (2.13) (az; q )~

is independent of a. Setting a = q we get

f ( q ) f ( O ) - (qz; q ) ~ " (2.14)

However, by (2.7),

~ - ~ ( q ; q ) s z ~ = ( 1 - z ) -1, since I z l < l . (2.15) f ( q ) = (q; q)~ s~0

Hence f ( 0 ) = 1/(1 - z ) ( q z ; q)~ = l / ( z ;q)~ , so we find that

(az; q )~ f ( a ) - - ( z ; q ) ~ ' (2.16)

which, in the standard notation, is the well-known q-binomial formula

I a ] _ (az;q)~ 1 (90 , q , z (z;~qT~ (2.17)

For a brief history o f this formula and a modern proof due to Askey (1980) see Gasper

and Rahman (1990).

3. q-linear lattice x(s ) = qS: Ramanujan's bilateral sum and integral

We now consider a slight extension of the results of the previous section by replacing

(2.1) by the following:

a(s) = (1 - bqS-1)q s, (3.~)

a(s) + z(s)Vxl(s) = z(1 - aqS)q s,

where b is an

we have

p(s + 1

additional parameter. When b = q this reduces to (2.1). By Eq. (1.14)

) 1 - aq s. , , -- ~q~tz/q; (3.2)

p(s) 1

whose solution is taken to be

(a;q)s p ( s ) - ( z / q ) ~, (3.3) (b;q)s

where the unit-periodic function is, once again, chosen to be 1. If b is not 1 or an integer power o f q, which we shall assume to be the case, then the denominator on

108 M. Rahman, S.K. Suslov/Journal of Stat&tical Planning and Inference 54 (1996) 101-118

the right side of (3.3) does not vanish for any integer s, positive or negative, with the understanding, o f course, that

( -q/a)nq("~) (a; q )_ , - , (3.4)

(q/a; q ) ,

see Gasper and Rahman (1990). Accordingly, our aim will be to evaluate the bilateral

s u m

( a ; q ) s s °~ ( a ; q ) s s ~--, ( q / b ; q ) s { b ) ~ g ( b ) : = Z - - z = Z - - z + L . . , ~, ) (3.5)

s = - ~ (b;q)s s=O (b;q)s s=l ( q / a ; q ) s -~z "

This sum is absolutely convergent in the annulus

- < [zl < 1, ( 3 . 6 ) a

which can be established by a simple ratio rest, see Gasper and Rahman (1990). Clearly, then, there is a finite region of analyticity of g(b) , as a function of b, around the point b -- 0. On the other hand, the series (3.5) is not always convergent at a = 0. This is the reason we intend to set up a recurrence relation in b instead of the numerator

parameter a as we did in Section 2. From (3.1) and (3.3) it is clear that

( a ; q ) ~ l i m z s = 0 if I z l < l (3.7) ~lina p ( s ) a ( s ) - (b; q ) ~ s ~

and s

s l im-o~P(s)~r(s) - (q /a ;q )o~ = 0 if < 1. (3.8)

So, the asymptotic condition (3.8) indicates a smaller region in the b-plane than is needed for the series in (3.5). This, however, is a minor problem which we shall be able to overcome in the end by appealing to an analytic continuation. By (1.13) and

(3.1) we get

A [p(s)~r(s)] = [z( 1 - aq s) - ( 1 - bq s - 1 )]p(s)qS

= - a z q b - l [ ( 1 - b / a q ) - (1 - b / a z q ) ( 1 - b q S - 1 ) ] p ( s ) q s. (3.9)

Since ( 1 - b q S - l ) / ( b ; q ) s = ( 1 - b / q ) / ( b / q ; q ) s , we find on using (3.5), (3.7) and (3.8) that

(1 - b / q ) ( 1 - b / a z q ) g ( b / q ) = (1 - b / a q ) g ( b )

which, on replacing b by bq, gives the recurrence relation

1 - b/a g ( b )

(1 - b ) (1 - b / a z ) 9(bq)" (3. 10)

M. Rahman, S.IC Suslov/Journal of Stat&tical Planning and Inference 54 (1996) 101 118 109

Iterating this n - 1 times and taking the limit n -~ oc we obtain

(b/a; q)~ g(b) - a(O). (3 .11)

(b, b/az; q)~

Since g(0) is independent of b we may set b = q in (3.1 1 ) to compute .q(0). However, by (3.5)

i (a;q)s s (az;q)~ g(q) = - - z - (3.12) ,=0 (q; q)" (z; q)~:

by (2.17). So

(q, q/az, az; q)~ g(0) = (3.13)

(z, q/a; q)

Substituting this in (3.11) we find the desired summation formula

(a;q)szs (q,b/a, az, q/az;q)o~ (3.14) , = - ~ ( b ; ~ = (q/a,b,z,b/az;q)~ "

This formula is certainly valid for all z satisfying Ib/aql < Iz[ < 1. But the series is absolutely convergent in the wider region (3.6) as are the infinite products on tile right side of (3.14). So, by analytic continuation (3.14) is valid in the annulus defined in (3.6). This is the famous 1~1 summation formula of Ramanujan, see Ramanujan, (1915) and Gasper and Rahman (1990), where r~kr is the bilateral sum defined by

[a,,a2 ar ] ~-~ (a,,a2 ar;q)nzn" . . . . . . . . . . rt~r Lbl,b2 ' ,b~ ;q, z j = . . . . (bl,b2, ~br;q)n (3.15)

There are many published proofs of this important formula. There is a close resem- blance between our proof and that given by Andrews and Askey (1978), For references to other proofs see Gasper and Rahman (1990).

Closely related to Ramanujan's sum is Ramanujan's integral, see Ramanujam (1915),

i_~ ~ h+s. ~oo F(a)F(1 - a~'(-q , ~ a)(qi ~,qb;q)~ • q ~ d s = l o ~ _ 5 (q, qb-,;q) ' (3.16)

where 0 < a < b. This can be proved in a very simple manner by using (3.14) and ob- serving (as was done in Ismail and Rahman, 1994) that if f ( x ) is a continuous function such that ~ _ ' _ ~ f ( x + n) converges uniformly on x E [0, 1] and f~,~ f ( x ) dx < oc, then

/~,C fo1 ~ f ( x ) do: = f ( x + n) dx. (3.1'7)

110 M. Rahman, S.K. SuslovlJournal of Statistical Plannm O and Inference 54 (1996) 101 118

Denoting the integral on the left side of (3.16) by I (a ,b) we find that

I . . . . f as(_q~.~;q)~ _qS (q, qb;q)~ t a ' ° ) - - I.So q 101 ;q, qa ds (q%qb_a;q) h(a),

(3.18)

where

/o' h(a) = ( -qa+~' -q l -a -s ;q )~qaS ds. (3.19) (__qS, _ql -s ; q )o~

To evaluate h(a) we notice

(qa, qb-a;q)~ /? h( a ) = q a" (-qb+'; q ) ~ ds (3.20) (q, qb;q)~ oo (--qS;q)oo

is independent of b, so we may take any special value of b such that the integral on the right side is bounded. The simplest such choice seems to be b = 1, since

I? as:S? - /0 q,~(-ql+S;q)o~ qa~ ds 1 ta- ldt _ F(a)F(1 - a)

oo (-q~',q)--~ o~ 1 T ~ logq -1 ] - T t logq -1

(3.21)

and hence

r(a)r(1 - a) (qa, ql-a; q)c~ h(a) = log q-1 (q, q; q ) ~ (3.22)

Combining (3.18) and (3.22) completes the proof of (3.16). Note that we need the condition 0 < a < 1 for the integral in (3.21) to converge, but we do not need it in (3.22) because of the factor (ql-a; q )~ . So, all we require for the final result to hold is the condition stated after (3.16). In fact, a and b can both be complex provided 0 < R e a < R e b .

The above proof of Ramanujan's integral is not new, rather a special case of the proof given in Rahman and Suslov (1994b) of Askey and Roy's (1986) extension of (3.16).

4. q-linear lattice x(s) = q-S: a Barnes-type integral

We shall now consider a Barnes-type integral where the integrand is a meromorphic function of q~ in the complex s-plane. There is no particular advantage of choosing the lattice to be q-S rather than qS, but the computations seem to be slightly easier. So we take

a(s) = (1 -- aq-S)q l-s,

a(s) + r (s )Vxl (s ) = t(q -s -- b)q-L (4.1)

M. Rahman, X K. Suslov / Journal of Statistical Planning and Inference 54 (1996) 10l 118 111

In this case

p ( s + 1) 1 ;qbq]~_ p(s) -- 1 - l ( tq-S)

whose general solution is

~(s) p ( ~ ) =

(aq -s, bqS; q ) ~

with

(4.2)

(4.3)

7r( s ) = ( aq ~, ql-S /a; q )ooqS (4.5)

so that n(s + l) /~(s) = (_q/e)q-S . Hence we must take e = -q / t . Thus

p(s) = ( - tq -S ' -qS+l / t ; q ) ~ (aq_S, bqS;q)oo q~. (4.6)

The integral we wish to evaluate is

J ( a , b ) : = f c p ( s ) q - ' d s = j l (-tq-S'-qS+l/t;q)°~dS,(aq -~, bqS; q ) ~ (4.7)

where C is the part o f the imaginary axis from - iTc/ logq - I to i~ / l ogq -1, which

corresponds to the lower and upper boundaries o f the basic rectangle of periodicity.

If labl < 1, which is what we shall assume to be the case, the integrand in (4.7) has no poles between C and C ' or on them, where C' is the contour described in

Section 1. In such a rectangle there can be no entire unit-periodic factor in (4.6) other

than a constant by virtue o f Liouville's theorem on doubly periodic functions, see, for example, Whittaker and Watson (1965, p. 431). So p(s) is the unique solution o f (4.2)

and is positive if b = a* and tt* -= q. Since ./c A[p(s)~r(s)] ds = 0 and

p(s )r (s )Vxl (s ) = [t(q " - b) - q(1 - aq-S)]p(s)q -s

t = - [ (1 - ab) - (1 + aq/t)(1 - aq ")lp(s)q -', (4.8)

a

we find that

1 + aq/t . . . . J ( a , b ) - l~_~-a~,aq, o). (4.9)

Iterating it n - 1 times we find that

(-aq/t; q), J(a, b) - J(aq n, b). (4.10)

(ab; q),

In order that p(s) depends on s only through qS and not t s we assume an ansatz:

7~(s + 1 ) - - tq -s. (4.4)

~(s)

112 M. Rahman, S.K. Suslov/Journal of Statistical Plannin 9 and Inference 54 (1996) 101-118

As n --~ cx~ no new singularities appear in the integrand of (4.7) after a is replaced by aq n. So we find that

J ( a , b ) - ( -aq / t ;q )~K(b) , (4.11) (ab; q)oo

where

K(b) = f c (-tq-S'-qS+l/t;q)°~ ds. (4.12) (bqS; q)o~

However,

(ab; q)o~ K(b) - J(a,b) (4.13)

(-aq/t; q )~

is independent o f a, so by setting a = - t in (4 .7 ) and (4 .13 ) we obtain

_ (-bt;q)oo f (-qs+l/t;q)o~ ds. K(b) (4.14) (q; q)o~ Jc (bqS; q)o~

To evaluate the integral on the right side of (4.14) let us first denote it by L(b) and consider another Pearson-type equation with x(s) = q~" and

a(s) = (1 + qS/t)q~, (4.15)

a(s) + z(s)Vxl(s) = (1 - bq~)q s,

so that po(s + 1)/po(s) = (1 -bqS) /q (1 + qs+l/t) with solution

po(s) - (-qs+a/t; q )~ (bqS;q)~ q-S (4.16)

Since

po(s)z(s)~7xl (s) = - ( b + 1/t)qSpo(s)q ~ = (1 + 1/bt)[1 - (1 - b~)]po(s)q s

and (1 - b q S ) p o ( s ) = (-qS+l/t; q)~/(bqS+l; q)~ we find that the corresponding recur- rence relation for L(b) is

L(b) = L(bq) (4.17)

which, on iteration, gives L(b) - -L (0 ) . Thus

K ( b ) - (~bt-;,q-)-)°~L(0) (4.18) tq;q)~

with

L(O) = f (-qS+l/t;q)°~ ds (4.19) (bqS; q)~ Jc

M. Rahman, S.I¢2 Suslov/Journal of Statistical Planning and Inference 54 (1996) 101 118 113

independent of b. We may therefore set b = -q/ t in (4.19) and find that L(0) is simply the length of C, i.e., 2n/logq -l. Hence, (4.11) and (4.19) give

J(a,b) (-aq/t;q)~ (-bt;q)oc . . . . 2hi (-aq/ t , -bt;q)~ . . . . r~u) = (4.20)

(ab; q)~ (q; q)oo " 1 o ~ --1 (q, ab; q)oo •

This proof is a bit long because we have chosen to use only the Pearson-type equations in two stages without borrowing any formulas from the basic hypergeometric function theory. It may be instructive to note that the proof could be considerably shortened had we chosen to use the 1~91 summation formula. Note that

-q/bt 1 (q, ab, -qS+l/t;q)~ itS1 -aq/t ;q'bq~ = -tq-S' (4.21 (-aq/t, -bt, aq -~, bqs; q )~ '

whose convergence on C requires only that ja] < 1 and ]b I < l. Substituting this in (4.7) we find that

( -aq/ t , -bt ;q)~ /.i~/logq 1 @ (-q/bt;q)n J(a, b) b~ q ~ ds

(q, ab;q)~ d_in/logq_ n=-~ (-aq/t;q),

(-aq/t,-bt;q)o~ ~ (-q/bt;q)nbn i / ~ (q, ab;q)~ ~= ~ (-aq/t;q)~ logq_ 1 ~ei~°d0 (4.22)

2hi (-aq/t, -bt; q)oo logq -1 (q, ab;q)~

This second approach was taken by Askey and Roy (1986) to give an extension of this formula which was later proved by Rahman and Suslov (1994b) by the first method.

Before closing this section we would like to point out that with the lattice as in (2 1 ), one might consider a solution of the form

s + l . p(s)q s = (q 'q)~zSn(s) (4.23) (aqS; q)~

where n(s + 1) = n(s) and integrate it from - i o o to ioc. Following Watson (1910) one would take n(s) = ei~'/sin ns. If Izl < 1 and larg ( - z ) l < n, then

f o o ds = (4.24) (qs+X;q)~ei~SzS 2i(q, az; q)~

i.~ (aqS; q)~ sin ns (a,z; q)~ '

which follows as a special case of Gasper and Rahman (1990, Eq. 4.2.2)), by virtue of the q-binomial formula.

114 M. Rahman, S.K. Suslov/Journal of Statistical Plannin9 and Inference 54 (1996) 101-118

5. q-linear lattice x ( s ) = qS: the 24~1 summation formulas

In this section we shall consider some extensions o f the q-b inomia l formula other

than the Ramanu jan sum. Accordingly , let us take

o-(s) -- (1 - ¢ ) ( 1 - c¢ -1 ) ,

a(s) + z(s)~Txl(s) = zq- l (1 - a ¢ ) ( 1 - b e ) . (5.1)

Then

"~(S)~7XI(S) = zq-~(1 - a ¢ ) ( 1 - b e ) - (1 - ¢ ' ) ( 1 - eqS-l ). (5.2)

Partial cancel la t ions on the right side o f (5.2) occur in two cases: ( I) z = c/ab; (II) z = q.

Case I. z = e/ab. In this case the coefficients o f q2S cancel out in (5.2) and we get

z(s)~Txl(s) = ¢ [ ( 1 - a ) (1 - e/aq) - (1 - c/abq)(1 - aq*)q-S]. (5.3)

The funct ion p(s) for a general z is

(a,b; aL p ( s ) - ( q , e ; ~ (z/q)~' (5.4)

and, in this part icular case, becomes

p(s) = p(s; a) - (a, b; q)~ (e/abq)S (5.5) (q, c; q)~

which leads to

A[p(s)a(s)] = (1 - a ) (1 - c/aq)p(s; a) - (1 - a ) (1 - c/abq)p(s; aq). (5.6)

However , p(s)a(s)]~=o----0 and

(a,b;q)o~ l im (c/abq) s = 0, (5.7) l i r n p(s)~r(s) - (q,e;q)o~ s ~ i f [e/abq] < l.

The series that we wish to evaluate is

L (a'b;q)s (c/ab) ~ f ( a ) :-- p(s)q ~ ---- (5.8) ~=0 ~=o (q ' e; q)~

which converges in the larger region ]c/ab[ < 1. So, an evaluat ion o f (5.8) for

]e/abq] < l will enable us to use an analyt ic cont inuat ion. By (5.6) and (5.8) we

then have

(l - c / a q ) f (a) ~- (1 - e/abq) f (aq) ,

which, on replacement o f a by a/q, becomes

1 - e/a f ( a ) -- f - -~ - ; f (a /q ) . (5.9)

c/at)

M. Rahman, S.K. Suslov/Journal of Statistical Planning and InJbrence 54 (1996) 101 118 115

F r o m (5.8) observe that

CX3 (b ;q ) s

lira f (aq -n) = Z " ~ ~=0 (q' c; q)~ q())(-c/b)s = f ( ~ ) (5 .10)

which a lways converges , see Gasper and R a h m a n (1990, p. 5). Thus, i terating (5.9)

n - 1 t imes and taking the l imit n --+ cx~ we obtain

f ( a ) - (c/a;q)o~ f (oc ) ( 5 . l l ) (c/ab; q)~

so that

(c/ab;q)~ (c/ab;q)~ ~-~ ( a , b ; ~ f ( ~ ) - (c/a;q)~ f ( a ) - (c/a;q)~ ~=0 (q,c; (c/ab)'" (5 .12)

must be independent o f a. But f ( 1 ) = 1, so sett ing a = 1 above we get f ( o c ) =

(c/b; q)~/(c; q)~ which, substi tuted in (5.11), g ives the so-cal led q -Gauss summat ion

formula

2~l [ a'cb ; q'c/ab ] - (c/a'c/b; c/ab; q),~ ' (5 .13)

wi th the restr ict ion that lc/abl < 1, instead o f the more restr ict ive one g iven in (5.7).

Case II. z = q. In this case the constant terms in (5.2) cancel out and we get

"c(s)~7Xl(S) : qS[(l -- a ) ( 1 - c/aq) - b(1 - c/abq)(1 - aqS)].

Also

so that

(5.14)

where

A [p(s)a(s)] = ( 1 - a)[ ( 1 - c/aq)p(s; q) - b( 1 - c/abq)p(s; aq)]. (5.16 )

Since a ( 0 ) = 0, we get p(s)a(s)ls=O = 0, but

(a,b;q)~ s l i m p(s)a(s) - (q, c; q)~" (5.1"7)

So, s u m m i n g the relat ion (5 .16) f rom s = 0 to s = cxD gives us a n o n h o m o g e n e o u s

recurrence formula

(aq, b; q)o~ - (1 - c/aq)gl(a) - b(1 - c/abq)gl(aq), (5.18

(q,c;q)oo

~ (a ,b ; q)s g l ( a ) = Z p ( s ) q s = Z ( q , c ; ~ q"

s = O s = O

(5 .19)

( a , b ; q)s p(s) = p(s;a) - - - (5 .15)

(q, c; q)s

116 M. Rahman, S.K. Suslov/Journal of Statistical Planning and Inference 54 (1996) 101 118

One could try to iterate (5.18) in the parameter a, or one could attempt at constructing another function that does satisfy a homogeneous recurrence. We will take this latter route. Note that c/ab remains unchanged if a, b, c are replaced by aq/c, bq/c and q2/c, respectively. Accordingly, we define

OQ

h(a) := ~ (aq/c, bq/c; q)s q,, s=0 (q' q2/c; q)"

and find that

( aq2 /c, bq/c; q )~ _ (1 - a)h(a ) _ bq( 1 _ c/abq)h(aq).

(q, q2/c; q)o~ a c

Let

(5.20)

(5.21)

c (c, aq/c, bq/c; q)o~ , , h ( a ) = - q ~ , ~ q ~ - ~ g2ta). (5.22)

Substituting it into (5.21) and simplifying the coefficients we obtain

(aq, b; q)~ - - (1 - c/aq)g2(a) + b(1 - c/abq)ga(aq). (5.23)

(q,c;q)o~

If we denote

g(a) = gl(a) + g2(a), (5.24)

then subtracting (5.23) from (5.18) gives

b(1 - c /~q)g(aq) _ 1 - abq/c g(a) 1 c/aq -1 Z ~q/c g(aq). (5.25)

Iterating it n - 1 times and then taking the limit n ~ ec we get

(abq/c; q)oo g(a) -- g(0). (5.26)

(aq/c; q)oc

It is clear from (5.19), (5.22) and (5.24) that g(0) := l i m n ~ g(aq n) exists. However,

(abq/c;(aq/c;q)~ (abq/c;(aq/c;q)~ 201[ ] g(0) - - q) g(a)-- ;q,q

q (a, b, q2/c; q)~ , - aq/c, bq/c ] c (abq/c, bq/c, c; q)~c q)l q2/c ; q, q J (5.27)

must be independent of a. So we may set a = c/q on the right side. This gives

q (c/q, q2/c; q)o~ (q/c; q)~ g(O) = c (c, bq/c; q)ec (bq/c; q)~" (5.28)

By (5.26) and (5.28), we have

(abq/c, q/c; q)~ g(a) = (aq/c, bq/c; q)o~ ' (5.29)

M. Rahman, S.K. Suslov/Journal of Statistical Planning and InJerence 54 (1996) 101 118 II 7

which amounts to the summation formula

2~, acb ;q,q - | ;q,q

(abq/c, q/c; q)~ = (5.30)

(aq/c, bq/c; q)~"

This is known as the nonterminating form of the q-Vandermonde sum, which was

proved in an entirely different way in Gasper and Rahman (1990). If a or b is a

negative integer power of q, say a --- q ", n = 0, 1,2 . . . . , then the second term on the

right side o f (5.30) vanishes, and we get

[ 7 I = (bql-%"c'q/c;q)~c -(c/b;q)"b", (5.31) q ,b ;q,q (bq/c,q 1 n,/c;q)~c ( c ; q ) , 2q~)l

which is one form of the q-Vandermonde (or q-Chu Vandermonde) sum, see Gasper

and Rahman (1990, Eq. (II.6)). By reversing its order of summation we get the other

form

(Gasper and Rahman, 1990, Eq. (II.7)), which, incidentally, follows from (5.13) if we set a = q-n.

6. Concluding remarks

An indefinite analogue o f the q-Gauss sum (5.13) was discussed in Rahman and

Suslov (1994b). But a full range of integration formulas, both on the real line and the

imaginary axis, requires a further extension o f the formulas given in Section 5, namely,

the bilateral 2@2 series. For reasons of space we postpone this to a future paper, where

we shall start with the 2~t2, continue on to 3~3, and conclude with a full treatment

of the summation and integration formulas arising out of the q-quadratic lattice. The

objective of this paper has been to show that there is a common link between the

various summation and integration formulas of basic hypergeometric series which have

been known for a long time, but whose original proofs did not seem to recognize this

connection.

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