Dieter Schmersau

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.

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 a structure formula for classical q -orthogonal polynomials

The classical orthogonal polynomials are given as the polynomial solutions Pn(x) of the di1erential equation � (x)y �� (x )+ � (x)y � (x )+ � ny(x )=0 ; where � (x) turns out to be a polynomial of at most second degree and � (x) is a polynomial of 5rst degree.In a similar way, the classical discrete orthogonal polynomials are the polynomial solutions of the di1erence equation � (x)7 y(x )+ � (x)7 y(x )+ � ny(x )=0 ; where 7 y(x )= y(x +1 )− y(x) and y(x )= y(x) − y(x − 1) denote the forward and backward di1erence operators, respectively.Finally, the classical q-orthogonal polynomials of the Hahn tableau are the polynomial solutions of the q-di1erence equation � (x)DqD1=qy(x )+ � (x)Dqy(x )+ � q; ny(x )=0 ; where Dqf(x )= f(qx) − f(x) (q − 1)x ;q � ; denotes the q-di1erence operator.We show by a purely algebraic deduction — without using the orthogonality of the families considered — that a structure formula of the type � (x)D1=qPn(x )= nPn+1(x )+ � nPn(x )+ � nPn−1(x )( n ∈ N:={1; 2; 3 ;::: }) is valid.Moreover, our approach does not only prove this assertion, but generates the form of this structure formula.A similar argument applies to the discrete and continuous cases and yields � (x)Pn(x )= nPn+1(x )+ � nPn(x )+ � nPn−1(x )( n ∈ N) and � (x)P �(x )= nPn+1(x )+ � nPn(x )+ � nPn−1(x )( n ∈ N):

Irrationality of certain infinite series

Abstract In this paper a new direct proof for the irrationality of Euler´s number e=∑k=0∞ 1/k! is presented. Furthermore, formulas for the base b digits are given which, however, are not computably effective. Finally we generalize our method and give a simple criterium for some fast converging series representing irrational numbers.

Irrationality of certain infinite series II

Abstract In a recent paper a new direct proof for the irrationality of Euler´s number e = ∑k = 0∞ 1/k! and on the same lines a simple criterion for some fast converging series representing irrational numbers was given. In the present paper, we give some generalizations of our previous results.