TY - GEN A1 - Koepf, Wolfram A1 - Schmersau, Dieter T1 - Algorithms for Classical Orthogonal Polynomials N2 - \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} T3 - ZIB-Report - SC-96-23 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2340 ER - TY - GEN A1 - Koepf, Wolfram T1 - Gröbner Bases and Triangles N2 - 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. T3 - ZIB-Report - SC-96-24 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2350 ER - TY - GEN A1 - Koepf, Wolfram A1 - Schmersau, Dieter T1 - Weinstein's Functions and the Askey-Gasper Identity N2 - \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. T3 - ZIB-Report - SC-96-06 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2175 ER - TY - GEN A1 - Koepf, Wolfram T1 - Efficient Computation of Orthogonal Polynomials in Computer Algebra N2 - 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. T3 - ZIB-Report - SC-95-42 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2080 ER - TY - GEN A1 - Koepf, Wolfram A1 - Schmersau, Dieter T1 - Representations of Orthogonal Polynomials N2 - {\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.} T3 - ZIB-Report - SC-97-06 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2756 ER - TY - GEN A1 - Koepf, Wolfram T1 - On a Problem of Koornwinder N2 - 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. T3 - ZIB-Report - SC-96-52 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2629 ER - TY - GEN A1 - Koepf, Wolfram T1 - A Package on Orthogonal Polynomials and Special Functions N2 - 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. T3 - ZIB-Report - SC-96-53 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2639 ER - TY - GEN A1 - Koepf, Wolfram T1 - Algorithms for the Indefinite and Definite Summation N2 - 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. T3 - ZIB-Report - SC-94-33 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1610 ER - TY - GEN A1 - Koepf, Wolfram T1 - Identities for Families of Orthogonal Polynomials and Special Functions N2 - 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. T3 - ZIB-Report - SC-95-01 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1675 ER - TY - GEN A1 - Koepf, Wolfram T1 - Algorithmic work with orthogonal polynomials and special functions. N2 - 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. T3 - ZIB-Report - SC-94-05 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1354 ER - TY - GEN A1 - Koepf, Wolfram T1 - A package on formal power series. N2 - 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.\ Stölting 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. T3 - ZIB-Report - SC-93-27 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1231 ER -