Wolfram Koepf

Representations of orthogonal polynomials

Abstract Zeilbergers algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computers 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 by 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.

On the Fekete-Szegö problem for close-to-convex functions II

In a previous paper [3] we solved the Fekete-Szeg6 problem of maximizing l a 3 - 2 a2l, 2 ~ [0, 1], for close-to-convex functions. The largest number 20 for which [a a - 20 a2[ is maximized by the Koebe function z/(1 - z) 2 is 20 = 1/3. Now we generalize this result to C (fl), fl > 1, showing that the largest number 20 (fl) for which l a 3 - 20 (fl)a2[ is maximized over C (fl) by k a with

Power series in computer algebra

Formal power series (FPS) of the form Σ k = 0 ∞ a k ( x − x 0 ) k are important in calculus and complex analysis. In some Computer Algebra Systems (CASs) it is possible to define an FPS by direct or recursive definition of its coefficients. Since some operations cannot be directly supported within the FPS domain, some systems generally convert FPS to finite truncated power series (TPS) for operations such as addition, multiplication, division, inversion and formal substitution. This results in a substantial loss of information. Since a goal of Computer Algebra is — in contrast to numerical programming — to work with formal objects and preserve such symbolic information, CAS should be able to use FPS when possible. There is a one-to-one correspondence between FPS with positive radius of convergence and corresponding analytic functions. It should be possible to automate conversion between these forms. Among CASs only M acsyma provides a procedure powerseries to calculate FPS from analytic expressions in certain special cases, but this is rather limited. Here we give an algorithmic approach for computing an FPS for a function from a very rich family of functions including all of the most prominent ones that can be found in mathematical dictionaries except those where the general coefficient depends on the Bernoulli, Euler, or Eulerian numbers. The algorithm has been implemented by the author and A. Rennoch in the CAS M athematica , and by D. Gruntz in M aple . Moreover, the same algorithm can sometimes be reversed to calculate a function that corresponds to a given FPS, in those cases when a certain type of ordinary differential equation can be solved.

Algorithms forq -Hypergeometric Summation in Computer Algebra

This paper describes three algorithms for q -hypergeometric summation: ? a multibasic analogue of Gosper?s algorithm, ? the q - Zeilberger algorithm, and ? an algorithm for finding q - hypergeometric solutions of linear recurrences together with their Maple implementations, which is relevant both to people being interested in symbolic computation and in q -series. For all these algorithms, the theoretical background is already known and has been described, so we give only short descriptions, and concentrate ourselves on introducing our corresponding Maple implementations by examples. Each section is closed with a description of the input/output specifications of the corresponding Maple command. We present applications to q -analogues of classical orthogonal polynomials. In particular, the connection coefficients between families of q -Askey?Wilson polynomials are computed. Mathematica implementations have been developed for most of these algorithms, whereas to our knowledge only Zeilberger?s algorithm has been implemented in Maple so far (Koornwinder, 1993 or Zeilberger, cf. Pe kov0sek et al., 1996). We made an effort to implement the algorithms as efficient as possible which in the q -Petkov?ek case led us to an approach with equivalence classes. Hence, our implementation is considerably faster than other ones. Furthermore the q -Gosper algorithm has been generalized to also find formal power series solutions.

IDENTITIES FOR FAMILIES OF ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONS

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. Nest, we present algorithms for holonomic families f(n x) of special functions which possess a derivative rule. We call those families admissible. A family f(n x) is holonomic if it satisfies a holonomic recurrence equation with repect to n, and a holonomic differential equation eith repect 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 of orthogonal polynomials, are admissible. We moreover present a method that generates ...

Fourth order q -difference equation for the first associated of the q -classical orthogonal polynomials

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 σ and τ which appear in the q-Pearson difference equation Dq(σ ϱ) = τϱ defining the weight ϱ of the q-classical orthogonal polynomials inside the q-Hahn tableau.

A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it

In this work, we present a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type and show that all the three classical orthogonal polynomial families as well as three finite orthogonal polynomial families, extracted from this equation, can be identified as special cases of this derived polynomial sequence. Some general properties of this sequence are also given.

Fourth-order difference equation for the associated classical discrete orthogonal polynomials

We derive the fourth-order dierence equation satised by the associated order r of classical orthogonal polynomials of a discrete variable. The coecients of this equation are given in terms of the polynomials and which appear in the discrete Pearson equation ()= dening the weight (x) of the classical discrete orthogonal polynomials. c 1998 Elsevier Science B.V. All rights reserved. AMS classication: 33C25

Bounded nonvanishing functions and bateman functions

We consider the family B of bounded nonvanishing analytic functions in the unit disk. The coefficient problem had been extensively investigated (see e.g. [2, 13, 14, 16-18, 20]), and it is known that for n = 1, 2, 3, and 4. That this inequality may hold for is known as the Krzyz conjecture. It turns out that for fΣ B with a 0=e -1 SO that the superordinate functions are of special interest. The corresponding coefficient functions F k(t) had been independently considered by Bateman [3] who had introduced them with the aid of the integral representation . We study the Bateman functions and formulate properties that give insight in the coefficient problem

On linearization and connection coefficients for generalized Hermite polynomials

Abstract We consider the problem of finding explicit formulas, recurrence relations and sign properties for both connection and linearization coefficients for generalized Hermite polynomials. Most of the computations are carried out by the computer algebra system Maple using appropriate algorithms.