arXiv subjects

Wolfram Koepf

Publications and source records attributed to Wolfram Koepf.

At least 19 recordsLinked to original sources

arXiv2022

FPS In Action: An Easy Way To Find Explicit Formulas For Interlaced Hypergeometric Sequences

Linear recurrence equations with constant coefficients define the power series coefficients of rational functions. However, one usually prefers to have an explicit formula for the sequence of coefficients, provided that such a formula is "simple" enough. Simplicity is related to the compactness of the formula due to the presence of algebraic numbers: "the smaller, the simpler". This poster showcases the capacity of recent updates on the Formal Power Series (FPS) algorithm, implemented in Maxima and Maple (convert/FormalPowerSeries), to find simple formulas for sequences like those from https://oeis.org/A307717, https://oeis.org/A226782, or https://oeis.org/A226784 by computing power series representations of their correctly guessed generating functions. We designed the algorithm for the more general context of univariate $P$-recursive sequences. Our implementations are available at http://www.mathematik.uni-kassel.de/~bteguia/FPS_webpage/FPS.htm

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arXiv2022

On the representation of non-holonomic univariate power series

Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important requirement to be satisfied by the function under consideration. The targeted functions mainly summarize that of meromorphic functions. However, expressions like $\tan(z)$, $z/(\exp(z)-1)$, $\sec(z)$, etc., particularly, reciprocals, quotients and compositions of holonomic functions, are generally not holonomic. Therefore their power series are inaccessible by the holonomic framework. From the mathematical dictionaries, one can observe that most of the known closed-form formulas of non-holonomic power series involve another sequence whose evaluation depends on some finite summations. In the case of $\tan(z)$ and $\sec(z)$ the corresponding sequences are the Bernoulli and Euler numbers, respectively. Thus providing a symbolic approach that yields complete representations when linear summations for power series coefficients of non-holonomic functions appear, might be seen as a step forward towards the representation of non-holonomic power series. By adapting the method of ansatz with undetermined coefficients, we build an algorithm that computes least-order quadratic differential equations with polynomial coefficients for a large class of non-holonomic functions. A differential equation resulting from this procedure is converted into a recurrence equation by applying the Cauchy product formula and rewriting powers into polynomials and derivatives into shifts. Finally, using enough initial values we are able to give normal form representations to characterize several non-holonomic power series and prove non-trivial identities. We discuss this algorithm and its implementation for Maple 2022.

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arXiv2021

Symbolic computation of hypergeometric type and non-holonomic power series

A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently, we describe an algorithm, say $mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis of the subspace of hypergeometric $(m=1)$ term solutions of holonomic recurrence equations to the more general case of $m$-fold hypergeometric terms. We generalize the concept of hypergeometric type power series introduced by Koepf (1992), by considering linear combinations of Laurent-Puiseux series whose coefficients are $m$-fold hypergeometric terms. Thus thanks to $mfoldHyper$, we deduce a complete procedure to compute these power series; indeed, it turns out that every linear combination of power series with $m$-fold hypergeometric term coefficients, for finitely many values of $m$, is detected. On the other hand, we investigate an algorithm to represent power series of non-holonomic functions. The algorithm follows the same steps of Koepf's algorithm, but instead of seeking holonomic differential equations, quadratic differential equations are computed and the Cauchy product rule is used to deduce recurrence equations for the power series coefficients. This algorithm defines a normal function that yields together with enough initial values normal forms for many power series of non-holonomic functions. Therefore, non-trivial identities are automatically proved using this approach. This paper is accompanied by implementations in the Computer Algebra Systems (CAS) Maxima 5.44.0 and Maple 2019.

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arXiv2018

Quasi-Orthogonality of Some Hypergeometric and $q$-Hypergeometric Polynomials

We show how to obtain linear combinations of polynomials in an orthogonal sequence $\{P_n\}_{n\geq 0}$, such as $Q_{n,k}(x)=\sum\limits_{i=0}^k a_{n,i}P_{n-i}(x)$, $a_{n,0}a_{n,k}\neq0$, that characterize quasi-orthogonal polynomials of order $k\le n-1$. The polynomials in the sequence $\{Q_{n,k}\}_{n\geq 0}$ are obtained from $P_{n}$, by making use of parameter shifts. We use an algorithmic approach to find these linear combinations for each family applicable and these equations are used to prove quasi-orthogonality of order $k$. We also determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the end points of the interval of orthogonality of the sequence $\{P_n\}_{n\geq 0}$, where possible.

math.CA↗

arXiv2014

Divisibility of Trinomials by Irreducible Polynomials over F2

Irreducible trinomials of given degree n over $F_2$ do not always exist and in the cases that there is no irreducible trinomial of degree n it may be effective to use trinomials with an irreducible factor of degree n. In this paper we consider some conditions under which irreducible polynomials divide trinomials over $F_2$. A condition for divisibility of self-reciprocal trinomials by irreducible polynomials over $F_2$ is established. And we extend Welch's criterion for testing if an irreducible polynomial divides trinomials $x^m+x^s+1$ to the trinomials $x^{am}+x^{bs}+1$.

math.RA↗

arXiv2013

Two Finite Classes of Orthogonal Functions

By using Fourier transforms of two symmetric sequences of finite orthogonal polynomials, we introduce two new classes of finite orthogonal functions and obtain their orthogonality relations via Parseval's identity.

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arXiv2004

Numerical algorithms for the real zeros of hypergeometric functions

Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying first order linear difference differential equations with continuous coefficients. In order to compute the zeros of arbitrary solutions of the hypergeometric equations, we have at our disposal several different sets of difference differential equations (DDE). We analyze the behavior of these different sets regarding the rate of convergence of the associated fixed point iteration. It is shown how combinations of different sets of DDEs, depending on the range of parameters and the dependent variable, is able to produce efficient methods for the computation of zeros with a fairly uniform convergence rate for each zero.

math.NA↗

arXiv1998

Software for the Algorithmic Work with Orthogonal Polynomials and Special Functions

In the last decade major steps towards an algorithmic treatment of orthogonal polynomials and special functions (OP & SF) have been made, notably Zeilberger's brilliant extension of Gosper's algorithm on algorithmic definite hypergeometric summation. By implementations of these and other algorithms symbolic computation has the potential to change the daily work of everybody who uses orthogonal polynomials or special functions in research or applications. It can be expected that symbolic computation will also play an important role in on-line versions of major revisions of existing formula books in the area of OP & SF. It this couple of talks I present software in Maple of those algorithmic techniques, in particular of Gosper's, Zeilberger's, and Petkovsek's algorithms and their q-analogoues. Some implementational details are discussed. The main emphasis, however, is given to on-line demonstrations of these algorithms using our Maple implementations (jointly with Harald Boeing) covering many examples from the field of OP & SF.

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arXiv1997

Representations of orthogonal polynomials

Zeilberger's algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computes recurrence and differential equations for hyperexponential integrals. Further versions of this algorithm allow the computation of recurrence and differential equations from Rodrigues type formulas and from generating functions. In particular, these algorithms can be used to compute the differential/difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations, and from their Rodrigues rperesentations and generating functions. In recent work, we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials, in terms of the coefficients of their differential/difference equations, to give an algorithm to identify the polynomial system from a given recurrence equation. In this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential/difference equation, the hupergeometric representation, and the recurrence equation is possible. The main technique is again to use texplicit formulas for structural identities of the given polynomial systems.

math.CA↗

arXiv1997

A package on orthogonal polynomials and special functions

In many applications (hupergeometric-type) special functions like orthogonal polynomials are needed. For example in more than 50% of the published solutions for the (application-oriented) questions in the "Problems Section" of SIAM Review special functions occur. In this article the Mathematica package SpecialFunction which can be obtained from the URL http://www.zib.de/koepf is introduced [15]. Algorithms to convert between power series representations and their generating functions is the main topic of this package {[8]-[15]}, extending the previous package PowerSeries [12]. Moreover the package automatically finds differential and recurrence equations {[13]-[14]} for expressions and for sums (the latter using Zeilberger's algorithm {[23], [18], [13\}. As an application the fast computation of polynomial approximations of solutions of linear differential equations with polynomial coefficients is presented. This is the asymptotically fastest known algorithm for series computations, and it is much faster than Mathematica's builtin Series command if applicable. Many more applications are considered. Finally the package includes implementations supporting the efficient computation of classical continuous and discrete orthogonal polynomials.

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arXiv1996

On a problem of Koornwinder

In this note we solve a problem about the rational representablility of hupergeometric terms which represent hypergeometric sums. This problem was proposed by Koornwinder in [4].

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arXiv1996

Algorithms for classical orthogonal polynomials

In this article explicit formulas for the recurrence equation p_{n+1}(x) = (A_n x + B_n) p_n(x) - C_n p_{n-1}(x) and the derivative rules sigma(x) p'_n(x) = alpha_n p_{n+1}(x) + beta_n p_n(x) + gamma_n p_{n-1}(x) and sigma(x) p'_n(x) = (alpha_n-tilde x + beta_n-tilde) p_n(x) + gamma_n-tilde p_{n-1}(x) respectively which are valid for the orthogonal polynomial solutions p_n(x) of the differential equation sigma(x) y''(x) + r(x) y'(x) + lambda_n y(x) = 0 of hypergeometric type are developed that depend only on the coefficients sigma(x) and tau(x) which themselves are polynomials w.r.t. x of degree not larger than 2 and 1, respectively. Partial solutions of this problem had beed previously published by Tricomi, and recently by Yáñez, Dehesa and Nikiforov.

math.CA↗

arXiv1996

The identification problem for transcendental functions

In this article algorithmic methods are presented that have essentially been introduced into computer algebra systems like Mathematica within the last decade. The main ideas are due to Stanley and Zeilberger. Some of them had already been discovered in the last century by Beke, but because of their complexity the underlying algorithms have fallen into oblivion. We combined these ideas, and added a factorization algorithm in noncommutative rings (Melenk--Koepf \cite{MK}) leading to a solution of the identification problem for a large class of transcendental functions. We present implementations of these algorithms in computer algebra systems.

math.CA↗

arXiv1996

On the De Branges theorem

Recently, Todorov and Wilf independently realized that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had introduced in his work. In this article, we present an elementary proof of this statement based on the defining differential equations system rather than the closed representation of de Branges' function system. Our proof does neither use special functions (like Wilf's) nor the residue theorem (like Todorov's) nor the closed representation (like both), but is purely algebraic. On the other hand, by a similar algebraic treatment, the closed representation of de Branges' function system is derived. In a final section, we give a simple representation of a generating function of the de Branges functions.

math.CA↗

arXiv1996

Weinstein's functions and the Askey-Gasper identity

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in 1976. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system which (by Todorov and Wilf) was realized to be the same as de Branges'. In this article, we show how a variant of the Askey-Gasper identity can be deduced by a straightforward examination of Weinstein's functions which intimately are related with a Löwner chain of the Koebe function, and therefore with univalent functions.

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arXiv1994

Algebraische Darstellung transzendenter Funktionen

Ich möchte in diesem Bericht algorithmische Methoden vorstellen, die im wesentlichen in diesem Jahrzehnt Einzug in die Computeralgebra gefunden haben. Die hauptsächlichen Ideen gehen auf Stanley \cite{Sta} und Zeilberger \cite{Zei1}--\cite{Zei4} zurück, vgl.\ die Beschreibung \cite{Strehl1}, und haben ihre Wurzeln teilweise bereits im letzten Jahrhundert (siehe z.\ B.\ \cite{Beke1}--\cite{Beke2}), gerieten aber auf Grund der Komplexität der auftretenden Algorithmen wieder in Vergessenheit.

math.CA↗

arXiv1994

Algorithms for the indefinite and definite summation

The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both $F(n+1,k)/F(n,k)$ and $F(n,k+1)/F(n,k)$ are rational functions. Typical examples are ratios of products of exponentials, factorials, $Γ$ function terms, bin omial coefficients, and Pochhammer symbols that are integer-linear with respect to $n$ and $k$ in their arguments. We consider the more general case of ratios of products of exponentials, factorials, $Γ$ function terms, binomial coefficients, and Pochhammer symbols that are rational-linear with respect to $n$ and $k$ in their arguments, and present an extended version of Zeilberger's algorithm for this case, using an extended version of Gosper's algorithm for indefinite summation. In a similar way the Wilf-Zeilberger method of rational function certification of integer-linear hypergeometric identities is extended to rational-linear hypergeometric identities.

math.CA↗

arXiv1994

REDUCE package for the indefinite and definite summation

This article describes the REDUCE package ZEILBERG implemented by Gregor Stölting and the author. The REDUCE package ZEILBERG is a careful implementation of the Gosper and Zeilberger algorithms for indefinite, and definite summation of hypergeometric terms, respectively. An expression $a_k$ is called a {\sl hypergeometric term} (or {\sl closed form}), if $a_{k}/a_{k-1}$ is a rational function with respect to $k$. Typical hypergeometric terms are ratios of products of powers, factorials, $Γ$ function terms, binomial coefficients, and shifted factorials (Pochhammer symbols) that are integer-linear in their arguments.

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