On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

Axioms

Mama Foupouagnigni¹  Wolfram Koepf²  Maurice Kenfack-Nangho¹ 

[1] Department of Mathematics, Higher Teachers’ Training College, University of Yaounde I, PO Box 47, Yaounde, Cameroon; E-Mail:;Institute of Mathematics, University of Kassel, Heinrich-Plett Street 40, Kassel 34132, Germany; E-Mail:

关键词: Askey-Wilson polynomials;    nonuniform lattices;    difference equations;    divided-difference equations;    Stieltjes function;   

DOI  :  10.3390/axioms2030404

来源: mdpi

【 摘 要 】

The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author.

【 授权许可】

CC BY   
© 2013 by the authors; licensee MDPI, Basel, Switzerland.