preliminary training: computer algebra, power series and ... · online demonstrations with computer...
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Preliminary Training:Computer Algebra,
Power Series and Summation
Prof. Dr. Wolfram Koepf
University of Kassel
www.mathematik.uni-kassel.de/˜koepf
AIMS-Volkswagen Stiftung Workshop onIntroduction to Orthogonal Polynomials and Applications
Preliminary Training 12October 6, 2018
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Online Demonstrations with Computer Algebra
Computer Algebra SystemsI will use the computer algebra system Maple to programand demonstrate the considered methods.Of course one could also easily use any other generalpurpose system like Mathematica, Maxima, Reduce orSage.
Needed AlgorithmsThe mostly used algorithms are:
Algorithms of Linear Algebra with many variables andcoefficients that are rational functions,multivariate polynomial factorization.
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Online Demonstrations with Computer Algebra
General-Purpose Computer Algebra Systemscontain a high level programming language,are dialog-oriented and not compiling,are able to plot functions,and can compute with symbols.
High-End AlgorithmsModular arithmetic,algebraic numbers,solving polynomial systems,differentiation and integration,solving differential equations,Taylor polynomials and power series.
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Example: Taylor Series
Robertson Conjecture I
The coefficients Bk (x) of the power series√√√√(1+z1−z
)x− 1
2xz=∞∑
k=0
Bk (x) zk
are polynomials of degree k .In a 1976 publication Malcolm S. Robertson conjecturedthat the coefficients of all these polynomials Bk (x) arenon-negative. He had computed Bk (x) for 1 5 k 5 6.This conjecture was repeated in an article published byRobertson in 1989.
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Example: Taylor Series
Robertson Conjecture IIIn this 1989 publication Robertson further conjectured thatthe coefficients Ak of the univariate power series√
ex − 1x
=∞∑
k=0
Ak xk
are all positive.It turns out that both conjectures are false!MapleBy these computations both conjectures are disproved.This result was published in 1991.Please notice that the reviewing process of my articlealmost 30 years ago was still very hard!
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Formal Power Series
Power Series RepresentationsSometimes one is not only interested in a Taylorpolynomial approximation, but in the full Taylor series.Assume, given an expression f (x) depending of thevariable x , we would therefore like to compute a formula forthe coefficient ak of the power series
f (x) =∞∑
k=0
ak xk
representing f (x).A well-known example of that type is given by
sin x =∞∑
k=0
(−1)k
(2k + 1)!x2k+1 .
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Formal Power Series
Algorithm FPS (Koepf, 1992)
Input: expression f (x)
HolonomicDE: Determine a holonomic differential equationDE (i.e. homogeneous and linear with polynomialcoefficients) by computing the derivatives of f (x) iteratively.DEtoRE: Convert DE to a holonomic recurrence equationRE for ak .RSolve: Solve RE for ak .
Output: ak resp.∞∑
k=0akxk
ExamplesMaple
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Computation of Holonomic Differential Equations
Algorithm HolonomicDE (Koepf, 1992)
Input: expression f (x)
Compute c0f (x) + c1f ′(x) + · · ·+ f (J)(x) with stillundetermined coefficients cj .Sort this linear combination w.r.t. linearly independentfunctions over Q(x) and determine their coefficients∈ Q(x).Set these coefficients zero, and solve the correspondinglinear system for the unknowns c0, c1, . . . , cJ−1.Output: DE := c0f (x) + c1f ′(x) + · · ·+ f (J)(x) = 0 (or elsemultiply by the common denominator of the cjs).
ExamplesMaple
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Algebraic Properties
Algebraic Perspective
The existence of a holonomic DE shows that thedimension of the vector space
Vf = 〈f (x), f ′(x), f ′′(x), . . .〉
over the field of rational functions Q(x) is finite.
Algorithms (Stanley, 1980), Maple (Salvy, Zimmermann, 1994)
Let a function f (x) be given by a holonomic differentialequation DE1 of order n, and let a function g(x) be givenby a holonomic differential equation DE2 of order m.Then there are linear algebra algorithms showing that f + gis holonomic of degree 5 n + m, and f · g is holonomic ofdegree 5 n ·m.Maple
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Holonomic Functions
Normal FormBy their algebraic properties the ring of holonomicfunctions has a so-called normal form which consists of thedifferential equation together with enough initial values.The same is true for the ring of holonomic sequenceswhose normal form consists of their recurrence equationtogether with enough initial values.By computing their normal forms two functions can beidentified as identical!MapleA function is holonomic iff it is the generating function of aholonomic sequence.Unfortunately, holonomic functions are not closed underdivision. Example: tan x = sin x
cos x .
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Hypergeometric Functions
Generalized Hypergeometric Series
A very rich class of holonomic functions are thehypergeometric functions / series.The formal power series
pFq
(a1, . . . ,ap
b1, . . . ,bq
∣∣∣∣∣ z)
=∞∑
k=0
Ak zk =∞∑
k=0
αk ,
whose summands αk = Akzk have a rational term ratio
αk+1
αk=
Ak+1 zk+1
Ak zk =(k + a1) · · · (k + ap)
(k + b1) · · · (k + bq)
z(k + 1)
,
is called the generalized hypergeometric series.
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Hypergeometric Functions
Hypergeometric Terms
For this purpose, the summand αk = Akzk of a hypergeometricseries is called a hypergeometric term.
Formula for Hypergeometric Terms
For the coefficients of the generalized hypergeometric functionone gets the following formula using the shifted factorial(Pochhammer symbol) (a)k = a(a + 1) · · · (a + k − 1)
pFq
(a1, . . . ,ap
b1, . . . ,bq
∣∣∣∣∣ z)
=∞∑
k=0
(a1)k · · · (ap)k
(b1)k · · · (bq)k
zk
k !.
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Hypergeometric Terms
First Order Holonomic Recurrence EquationsTherefore every holonomic recurrence equation of firstorder defines a hypergeometric term which can bedetermined by the hypergeometric coefficient formula.If the recurrence is brought into the form
αk+1
αk=
(k + a1) · · · (k + ap)
(k + b1) · · · (k + bq)
z(k + 1)
,
or, equivalently
(k+b1) · · · (k+bq)(k+1)αk+1−(k+a1) · · · (k+ap) z αk = 0 ,
then αk is given by
αk =(a1)k · · · (ap)k
(b1)k · · · (bq)k
zk
k !·α0 .
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Identification of Hypergeometric Functions / Terms
Algorithm Sumtohyper (Koepf 1998)The above method provides an algorithm to detect thehypergeometric representation for
S =∞∑
k=0
αk .
Input: αk
Computerk :=
αk+1
αk∈ Q(k) .
Factorize rk .Output: Read off upper and lower parameters, theargument z, and compute an initial value.
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Identification of Hypergeometric Functions / Terms
ExampleFor
sin x =∞∑
k=0
αk =∞∑
k=0
(−1)k
(2k + 1)!x2k+1 ,
we get
rk =αk+1
αk= − x2
2(k + 1)(2k + 3)∈ Q(x , k)
andα0 = x .
Thereforesin x = x · 0F1
(−
3/2
∣∣∣∣∣−x2
4
).
Maple
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Fasenmyer’s Algorithm (1949)
Binomial sumsThe sum evaluation
n∑k=0
(nk
)= 2n
is well-known.However, the similar, but more advanced identities
n∑k=0
k(
nk
)= n 2n−1 and
n∑k=0
(nk
)2
=
(2nn
)might be less known.Our question is: How can we compute the right hand sides,given the sums on the left?Maple
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Fasenmyer’s Algorithm (1949)
Applying k -free recurrence for the summand
Consider the sum
Sn :=n∑
k=0
F (n, k) withF (n+1, k)
F (n, k),F (n, k +1)
F (n, k)∈ Q(n, k).
F (n, k) is a hypergeometric term w r. t. k and n.Fasenmyer’s idea is to find first a k -free recurrence for thesummand F (n, k):
J∑j=0
I∑i=0
aij F (n + j , k + i) = 0 for some aij ∈ Q(n) .
This can be done by linear algebra!Since the recurrence is k -free, it can be summed fork = −∞, . . . ,∞ to get a pure recurrence for the sum Sn.
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Fasenmyer’s Algorithm (1949)
ExampleFor
F (n, k) =
(nk
)and Sn =
n∑k=0
(nk
)=
∞∑k=−∞
(nk
)the first step yields the Pascal triangle identity
F (n + 1, k + 1) = F (n, k + 1) + F (n, k) .
If we sum this relation for k = −∞, . . . ,∞, we clearly get
Sn+1 = Sn + Sn = 2 Sn .
Since S0 = 1, we finally have Sn = 2n.Maple
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Gosper’s Algorithm (1978)
AntidifferenceGiven ak , a sequence sk is called an antidifference of ak if
ak = sk+1 − sk .
Given such an antidifference, then by telescoping the sumcan be computed with arbitrary lower and upper bounds:
n∑k=m
ak = (sn+1 − sn) + (sn − sn−1) + · · ·+ (sm+1 − sm)
= sn+1 − sm .
Gosper’s algorithm computes a hypergeometric termantidfference sk for a hypergeometric term ak , i.e.
ak+1
ak∈ Q(k) .
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Gosper’s Algorithm (1978)
Computational stepsGiven ak with ak+1
ak=
uk
vk
with polynomials uk , vk ∈ Q[k ], it is easy to show that
sk =gk
hkak
for certain polynomials gk ,hk ∈ Q[k ].Whereas the denominator hk can be written down explicitly,for gk one gets the inhomogeneous recurrence equation
hk uk gk+1 − hk+1 vk gk = hk hk+1 vk .
To compute gk one first finds a degree bound for gk .The final step uses linear algebra to compute thecoefficients of gk by equating coefficients.
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Gosper’s Algorithm (1978)
Gosper’s implementationGosper’s algorithm was maybe the first algorithm whichwould not have been found without computer algebra. Inhis paper Gosper stated:Without the support of MACSYMA and its developers, Icould not have collected the experiences necessary toprovoke the conjectures that led to this algorithm.
Examples
An antidifference of ak = (−1)k (nk
)is sk = − k
n ak .
The harmonic numbers Hn =∑n
k=11k cannot be written as
hypergeometric term.Maple
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Zeilberger’s Algorithm (1990)
Applying Gosper’s algorithm successfully to definite sumsZeilberger’s algorithm deals again—like Fasenmyer’s—withdefinite sums of the form
Sn :=n∑
k=0
F (n, k) withF (n+1, k)
F (n, k),F (n, k +1)
F (n, k)∈ Q(n, k).
Applying Gosper to F (n, k) w. r. t. k (by telescoping)always generates the identity Sn = 0.Therefore this algorithm generally does not work.Doron Zeilberger’s idea: Apply Gosper instead to
ak := F (n, k) +J∑
j=1
σj F (n + j , k) for some J = 1 .
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Zeilberger’s Algorithm (1990)
Applying Gosper’s algorithm successfully to definite sums
Doron Zeilberger’s idea: Apply Gosper instead to
ak := F (n, k) +J∑
j=1
σj F (n + j , k) for some J = 1 .
In the final linear algebra step, solve not only for thecoefficients of gk , but also for σj , (j = 1, . . . , J).If successful, then this algorithm generates the recurrenceequation
Sn +J∑
j=1
σj Sn+j = 0
with rational functions σj ∈ Q(n), (j = 1, . . . , J) for Sn.Multiplication with the common denominator yields aholonomic recurrence equation for Sn.
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Doron Zeilberger
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Online Orthogonal Polynomials
CAOP = Computer Algebra and Orthogonal Polynomials
CAOP is a web tool for calculating formulas for orthogonalpolynomials belonging to the Askey-Wilson scheme usingMaple.The implementation of CAOP was originally done by RenéSwarttouw as part of the Askey-Wilson Scheme Projectperformed at RIACA in Eindhoven in 2004.The present site is a completely revised version of thisproject which has been done by Torsten Sprenger undermy supervision in 2012 and is maintained at the Universityof Kassel.http://www.caop.org/
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Petkovšek’s Algorithm (1992)
Hypergeometric solutions of holonomic recurrences
Assume, Sn is a hypergeometric term, howeverZeilberger’s algorithm generates not the recurrence of firstorder, but a holonomic recurrence of higher order
RE :J∑
j=0
Pj(n) Sn+j = 0
with polynomial coefficients Pj ∈ Q[n].Then Petkovšek’s algorithm—which is quite similar toGosper’s—finds every solution which is a linearcombination of hypergeometric terms.Maple
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
Van Hoeij’s Algorithm (1999)
Hypergeometric solutions of holonomic recurrences
Combining Zeilberger’s with Petkovšek’s algorithm leads toa decision procedure for hypergemetric term summation.However, Petkovšek’s algorithm is rather inefficient.Mark van Hoeij gave a much faster algorithm for the samepurpose, based on the singularity behavior of theconstituting Γ functions that occur in the solutions.Van Hoeij’s algorithm is implemented in Maple. Details canbe found in my book.Maple
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CA FPS – Holonomic Algebra Hypergeometrics Fasenmyer Gosper Zeilberger Petkovšek/van Hoeij Finale
I would like to thank you very much for your interest!