Axioms 2013, 2(3), 404-434; doi:10.3390/axioms2030404

On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

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Received: 28 May 2013; in revised form: 4 June 2013 / Accepted: 3 July 2013 / Published: 23 July 2013
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract: 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.
Keywords: Askey-Wilson polynomials; nonuniform lattices; difference equations; divided-difference equations; Stieltjes function
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MDPI and ACS Style

Foupouagnigni, M.; Koepf, W.; Kenfack-Nangho, M.; Mboutngam, S. On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices. Axioms 2013, 2, 404-434.

AMA Style

Foupouagnigni M, Koepf W, Kenfack-Nangho M, Mboutngam S. On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices. Axioms. 2013; 2(3):404-434.

Chicago/Turabian Style

Foupouagnigni, Mama; Koepf, Wolfram; Kenfack-Nangho, Maurice; Mboutngam, Salifou. 2013. "On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices." Axioms 2, no. 3: 404-434.

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