@misc{BoeingKoepf, author = {B{\"o}ing, Harald and Koepf, Wolfram}, title = {REDUCE Package for the Indefinite and Definite Summation of q-hypergeometric Terms}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5460}, number = {TR-97-04}, abstract = {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.}, language = {en} } @misc{KoepfSchmersau, author = {Koepf, Wolfram and Schmersau, Dieter}, title = {Algorithms for Classical Orthogonal Polynomials}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2340}, number = {SC-96-23}, abstract = {\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}}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {Gr{\"o}bner Bases and Triangles}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2350}, number = {SC-96-24}, abstract = {It is well-known that by polynomial elimination methods, in particular by the computation of Gr{\"o}bner bases, proofs for geometric theorems can be automatically generated. \%\% Several monographs On the other hand, it is much less known that Gr{\"o}bner 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.}, language = {en} } @misc{KoepfSchmersau, author = {Koepf, Wolfram and Schmersau, Dieter}, title = {Weinstein's Functions and the Askey-Gasper Identity}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2175}, number = {SC-96-06}, abstract = {\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 L{\"o}wner 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 L{\"o}wner chain of the Koebe function, and therefore with univalent functions.}, language = {en} } @misc{KoepfSchmersau, author = {Koepf, Wolfram and Schmersau, Dieter}, title = {Spaces of Functions Satisfying Simple Differential Equations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5074}, number = {TR-94-02}, abstract = {{\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.}}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {On Families of Iterated Derivatives.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5083}, number = {TR-94-03}, abstract = {We give an overview of an approach on special functions due to Truesdell, and show how it can be used to develop certain type of identities for special functions. Once obtained, these identities may be verified by an independent algorithmic method for which we give some examples.}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {REDUCE Package for the Indefinite and Definite Summation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5147}, number = {TR-94-09}, abstract = {This article describes the REDUCE package ZEILBERG implemented by Gregor St{\"o}lting 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.}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {REDUCE Packages on Power Series, Z-Transformation, Residues and Trigonometric Simplification}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5187}, number = {TR-95-03}, abstract = {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.}, language = {en} } @misc{KoepfSchmersau, author = {Koepf, Wolfram and Schmersau, Dieter}, title = {Representations of Orthogonal Polynomials}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2756}, number = {SC-97-06}, abstract = {{\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.}}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {Algorithms for the Indefinite and Definite Summation}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1610}, number = {SC-94-33}, abstract = {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.}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {Identities for Families of Orthogonal Polynomials and Special Functions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1675}, number = {SC-95-01}, abstract = {In this article we present new results for families of orthogonal polynomials and special functions, that are determined by algorithmical approaches. In the first section, we present new results, especially for discrete families of orthogonal polynomials, obtained by an application of the celebrated Zeilberger algorithm. Next, we present algorithms for holonomic families \$f(n,x)\$ of special functions which possess a derivative rule. We call those families {\sl admissible}. A family \$f(n,x)\$ is holonomic if it satisfies a holonomic recurrence equation with respect to \$n\$, and a holonomic differential equation with respect to \$x\$, i.\ e. linear homogeneous equations with polynomial coefficients. The rather rigid property of admissibility has many interesting consequences, that can be used to generate and verify identities for these functions by linear algebra techniques. On the other hand, many families of special functions, in particular families of orthogonal polynomials, are admissible. We moreover present a method that generates the derivative rule from the holonomic representation of a holonomic family. \% whenever one exists. As examples, we find new identities for the Jacobi polynomials and for the Whittaker functions, and for families of discrete orthogonal polynomials by the given approach. Finally, we present representations for the parameter derivatives of the Gegenbauer and the generalized Laguerre polynomials.}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {Algorithmic work with orthogonal polynomials and special functions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1354}, number = {SC-94-05}, abstract = {In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types of identities for those functions. Some of these identities like differential equations, power series representations, and hypergeometric representations can even dealt with algorithmically, i.\ e.\ they can be computed by the Computer Algebra system, rather than only verified. The types of functions that can be treated by the given technique cover the generalized hypergeometric functions, and therefore most of the special functions that can be found in mathematical dictionaries. The types of identities for which we present verification algorithms cover differential equations, power series representations, identities of the Rodrigues type, hypergeometric representations, and algorithms containing symbolic sums. The current implementations of special functions in existing Computer Algebra systems do not meet these high standards as we shall show in examples. They should be modified, and we show results of our implementations.}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {A package on formal power series.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1231}, number = {SC-93-27}, abstract = {Formal Laurent-Puiseux series of the form \[ f(x)=\sum \limits_{k=k_0}^{\infty}a_{k}x^{k/n} \] are important in many branches of mathematics. Whereas {\sc Mathematica} supports the calculation of truncated series with its {\tt Series} command, and the {\sc Mathematica} package {\tt SymbolicSum} that is shipped with {\sc Mathematica} version 2 is able to convert formal series of the type mentioned above in some instances to their corresponding generating functions, in six publications of the author we developed an algorithmic procedure to do these conversions that is implemented by the author, A.\ Rennoch and G.\ St{\"o}lting in the {\sc Mathematica} package {\tt PowerSeries}. The implementation enables the user to reproduce most of the results of the extensive bibliography on series of Hansen, E.\ R.: A table of series and products. Prentice-Hall, 1975. Moreover a subalgorithm of its own significance generates differential equations satisfied by the input function.}, language = {en} } @misc{GruntzKoepf, author = {Gruntz, Dominik and Koepf, Wolfram}, title = {Formal Power Series.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1267}, number = {SC-93-31}, abstract = {{\newcommand{\C}{{\rm {\mbox{C{\llap{{\vrule height1.52ex}\kern.4em}}}}}} \newcommand{\Z} {{\rm {\mbox{\protect\makebox[.2em][l]{\sf Z}\sf Z}}}} \newcommand{\Maple}{{\sc Maple}} Formal Laurent-Puiseux series of the form \[ f(x)=\sum\limits_{k=k_0}^{\infty}a_{k}x^{k/n} \label{eq:formalLPS} \] with coefficients \$a_{k}\in\C\;(k\in\Z)\$ are important in many branches of mathematics. \Maple\ supports the computation of {\em truncated\/} series with its {\tt series} command, and through the {\tt powerseries} package infinite series are available. In the latter case, the series is represented as a table of coefficients that have already been determined together with a function for computing additional coefficients. This is known as {\em lazy evaluation\/}. But these tools fail, if one is interested in an explicit formula for the coefficients \$a_k\$. In this article we will describe the \Maple\ implementation of an algorithm presented in several papers of the second author which computes an {\em exact\/} formal power series of a given function. This procedure will enable the user to reproduce most of the results of the extensive bibliography on series. We will give an overview of the algorithm and then present some parts of it in more detail. This package is available through the \Maple-share library with the name {\tt FPS}. We flavor this procedure with the following example. \%\begin{maple} \begin{verbatim}> FormalPowerSeries(sin(x), x=0);\end{verbatim} \begin{samepage} \begin{verbatim} infinity ----- k (2 k + 1) \ (-1) x ) ---------------- / (2 k + 1)! ----- k = 0 \end{verbatim} \end{samepage} }}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {Algebraische Darstellung transzendenter Funktionen}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1524}, number = {SC-94-24}, abstract = {In this article algorithmic methods are presented that have been incorporated in computer algebra systems in the last five years. The methods can be used to identify transcendental functions. The essential idea due to Zeilberger is to represent the functions by systems of holonomic differential and recurrence equations, and the identification procedure utilizes algorithms of Stanley and Zeilberger. Also a noncommutative version of the Buchberger algorithm can be applied.}, language = {de} } @misc{KoepfSchmersau, author = {Koepf, Wolfram and Schmersau, Dieter}, title = {On the De Branges Theorem}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1765}, number = {SC-95-10}, abstract = {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 L{\"o}wner differential equation, plus properties of the Koebe function.}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {The Identification Problem for Transcendental Functions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1791}, number = {SC-95-13}, abstract = {In this article algorithmic methods are presented that have essentially been introduced into computer algebra within the last decade. The main ideas are due to Stanley[1980] and Zeilberger[1990]. Some of them had already been discovered in the last century (see e.\ g.\ Beke[1894]), but because of the complexity of the underlying algorithms have fallen into oblivion. The combination of these ideas leads to a solution of the identification problem for a large class of transcendental functions. We present implementations of these algorithms in computer algebra systems.}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {Efficient Computation of Orthogonal Polynomials in Computer Algebra}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2080}, number = {SC-95-42}, abstract = {Orthogonal polynomials \%like the Chebyshev polynomials can be calculated by computation of determinants, by the use of generating functions, in terms of Rodrigues formulas, by iterating recurrence equations, calculating the polynomial solutions of differential equations, through closed form representations and by other means. In this article, we give an overview about the efficiency of the above methods in Maple, Mathematica, and REDUCE. As a noncommercial package we include the MuPAD system.}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {On a Problem of Koornwinder}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2629}, number = {SC-96-52}, abstract = {In this note we solve a problem about the rational representability of hypergeometric terms which represent hypergeometric sums. This problem was proposed by Koornwinder in Koornwinder, T. H.: Hypergeometric series evaluation by Zeilberger's algorithm. In: Open Problems, ed. by Walter van Assche. J. of Comput. and Appl. Math.48, 1993, 225--243.}, language = {en} } @misc{Koepf, author = {Koepf, Wolfram}, title = {A Package on Orthogonal Polynomials and Special Functions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2639}, number = {SC-96-53}, abstract = {In many applications (hypergeometric-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 {\tt SpecialFunctions} which can be obtained from the URL {\tt http://www.zib.de/koepf} is introduced. Algorithms to convert between power series representations and their generating functions is the main topic of this package, extending the previous package {\tt PowerSeries}. Moreover the package automatically finds differential and recurrence equations for expressions and for sums (the latter using Zeilberger's algorithm. 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 {\tt 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.}, language = {en} } @misc{FoupouagnigniKoepfRonveaux, author = {Foupouagnigni, Mama and Koepf, Wolfram and Ronveaux, Andre}, title = {Fourth Order Difference Equation for the Associated Classical Discrete Orthogonal Polynomials}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3419}, number = {SC-97-72}, abstract = {We derive the fourth order difference equation satisfied by the associated of order \$\;r\;\$ of the classical orthogonal polynomials of a discrete variable.\\The coefficients of this equation are given in terms of the polynomials \$\; \sigma\;\$ and \$\;\tau\;\$ which appear in the discrete Pearson equation \$\;\;\Delta(\sigma\;\rho)=\tau\;\rho\;\;\$ defining the weight \$\;\rho(x)\;\$ of the classical discrete orthogonal polynomials.}, language = {en} } @misc{BoeingKoepf, author = {B{\"o}ing, Harald and Koepf, Wolfram}, title = {Algorithms for q-hypergeometric Summation in Computer Algebra}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3453}, number = {SC-98-02}, abstract = {In this paper we present a short description of \$q\$-analogues of Gosper's, Zeilberger's, Petkov\v{s}ek's and related algorithms. Furthermore we introduce our corresponding MAPLE implementations and show how they can be applied to prove or even derive identities associated with \$q\$-series.}, language = {en} } @misc{FoupouagnigniRonveauxKoepf, author = {Foupouagnigni, Mama and Ronveaux, Andre and Koepf, Wolfram}, title = {Fourth-Order q-Difference Equation for the First Associated of the q-Classical Orthogonal Polynomials}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3490}, number = {SC-98-06}, abstract = {We derive the fourth order \$q\$-difference equation satisfied by the first associated of the \$q\$-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials \$\; \sigma\;\$ and \$\;\tau\;\$ which appear in the \$q\$-Pearson difference equation \$\;\; D_q(\sigma\,\rho)=\tau\,\rho\;\$ defining the weight \$\rho\$ of the \$q\$-classical orthogonal polynomials inside the \$q\$-Hahn tableau.}, language = {en} }