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This package is an implementation of the $q$-analogues of Gosper's and Zeilberger's algorithm for indefinite, and definite summation of $q$-hypergeometric terms, respectively. An expression $a_k$ is called a {\sl $q$-hypergeometric term}, if $a_{k}/a_{k-1}$ is a rational function with respect to $q^k$. Most $q$-terms are based on the {\sl $q$-shifted factorial} or {\sl qpochhammer}. Other typical $q$-hypergeometric terms are ratios of products of powers, $q$-factorials, $q$-binomial coefficients, and $q$-shifted factorials that are integer-linear in their arguments.
\begin{enumerate} \item[] {{\small 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 \[ \sigma(x)\,p_n'(x)=(\tilde\alpha_n\,x+\tilde\beta_n)\,p_n(x)+ \tilde\gamma_n\,p_{n-1}(x) \] respectively which are valid for the orthogonal polynomial solutions $p_n(x)$ of the differential equation \[ \sigma(x)\,y''(x)+\tau(x)\,y'(x)+\lambda_n\,y(x)=0 \] of hypergeometric type are developed that depend {\sl only} on the coefficients $\sigma(x)$ and $\tau(x)$ % and $\lambda_n$ which themselves are polynomials w.r.t.\ $x$ of degrees not larger than $2$ and $1$% and $0$ , respectively. Partial solutions of this problem had been previously published by Tricomi, and recently by Y\'a\~nez, Dehesa and Nikiforov. Our formulas yield an algorithm with which it can be decided whether a given holonomic recurrence equation (i.e.\ one with polynomial coefficients) generates a family of classical orthogonal polynomials, and returns the corresponding data (density function, interval) including the standardization data in the affirmative case. In a similar way, explicit formulas for the coefficients of the recurrence equation and the difference rule \[ \sigma(x)\,\nabla p_n(x)= \alpha_n\,p_{n+1}(x)+\beta_n\,p_n(x)+\gamma_n\,p_{n-1}(x) \] of the classical orthogonal polynomials of a discrete variable are given that depend only on the coefficients $\sigma(x)$ and $\tau(x)$ of their difference equation \[ \sigma(x)\,\Delta\nabla y(x)+\tau(x)\,\Delta y(x)+\lambda_n\,y(x)=0 \;. \] Here \[ \Delta y(x)=y(x+1)-y(x) \quad\quad\mbox{and}\quad\quad \nabla y(x)=y(x)-y(x-1) \] denote the forward and backward difference operators, respectively. In particular this solves the corresponding inverse problem to find the classical discrete orthogonal polynomial solutions of a given holonomic recurrence equation. \iffalse Furthermore, an algorithmic approach to deduce these and similar properties is presented which is implementable in computer algebra, and which moreover generates relations between different standardizations of the polynomial system considered. \fi }} \end{enumerate}
Gröbner Bases and Triangles
(1996)
It is well-known that by polynomial elimination methods, in particular by the computation of Gröbner bases, proofs for geometric theorems can be automatically generated. %% Several monographs On the other hand, it is much less known that Gröbner bases, in combination with rational factorization, can be even used to {\sl find} new geometric theorems. In this article such a method is described, and some new theorems on plane triangles are deduced.
\iffalse 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. Our whole contribution can be looked at as the study of properties of the Koebe function. Therefore, in a very elementary manner it is shown that the known proofs of the Bieberbach and Milin conjectures can be understood as a consequence of the Löwner differential equation, plus properties of the Koebe function. \fi 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.
{\newcommand{\N} {{\rm {\mbox{\protect\makebox[.15em][l]{I}N}}}} In several publications the first author published an algorithm for the conversion of analytic functions for which derivative rules are given into their representing power series $\sum\limits_{k=0}^{\infty}a_{k}z^{k}$ at the origin and vice versa, implementations of which exist in {\sc Mathematica}, {\sc Maple} and {\sc Reduce}. One main part of this procedure is an algorithm to derive a homogeneous linear differential equation with polynomial coefficients for the given function. We call this type of ordinary differential equations {\sl simple}. Whereas the opposite question to find functions satisfying given differential equations is studied in great detail, our question to find differential equations that are satisfied by given functions seems to be rarely posed. In this paper we consider the family $F$ of functions satisfying a simple differential equation generated by the rational, the algebraic, and certain transcendental functions. It turns out that $F$ forms a linear space of transcendental functions. % with polynomial function coefficients. Further $F$ is closed under multiplication and under the composition with rational functions and rational powers. These results had been published by Stanley who had proved them by theoretical algebraic considerations. In contrast our treatment is purely algorithmically oriented. We present algorithms that generate simple differential equation for $f+g$, $f\cdot g$, $f\circ r$ ($r$ rational), and $f\circ x^{p/q}$ ($p,q\in\N_0$), given simple differential equations for $f$, and $g$, and give a priori estimates for the order of the resulting differential equations. We show that all order estimates are sharp. After finishing this article we realized that in independent work Salvy and Zimmermann published similar algorithms. Our treatment gives a detailed description of those algorithms and their validity.}
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 (The sum package contains also a partial implementation of the Gosper algorithm.) 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, $\Gamma$ function terms, binomial coefficients, and shifted factorials (Pochhammer symbols) that are integer-linear in their arguments.
In this report, we present a collection of new REDUCE packages that recently have been developed. These are the packages {\tt FPS}, {\tt ZTRANS}, {\tt RESIDUE} and {\tt TRIGSIMP} on the following topics: \begin{itemize} \item[] FPS Calculation of differential equations and formal power series representations, in particular for orthogonal polynomials and special functions of the hypergeometric type. \item[] ZTRANS Calculation of the Z-Transformation and its inverse. % to be used for the solution of linear recurrence equations. \item[] RESIDUE Calculation of residues of meromorphic functions. \item[] TRIGSIMP Simplification of expressions involving trigonometric and hyperbolic functions. \end{itemize} % These packages can be obtained via anonymous {\tt ftp} from % {\tt } in the directories {\tt ..}, ) For each of these packages, a description in form of a \LaTeX\ file is distributed together with the package. These documentations are collected here. In a final chapter, we show how to solve some difficult problems with these packages, and how they usefully can be combined.
{\small 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 representations 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 be presenting a collection of algorithms with which any of the conversions between the differential/difference equation, the hypergeometric representation, and the recurrence equation is possible. The main technique is again to use explicit formulas for structural identities of the given polynomial systems.}
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 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, $\Gamma$ function terms, binomial 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, $\Gamma$ 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. The given algorithms on definite summation apply to many cases in the literature to which neither the Zeilberger approach nor the Wilf-Zeilberger method is applicable. Examples of this type are given by theorems of Watson and Whipple, and a large list of identities (``Strange evaluations of hypergeometric series'') that were studied by Gessel and Stanton. It turns out that with our extended algorithms practically all hypergeometric identities in the literature can be verified. Finally we show how the algorithms can be used to generate new identities. REDUCE and MAPLE implementations of the given algorithms can be obtained from the author, many results of which are presented in the paper.