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

On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

1,* , 2
,
1
 and
3
1 Department of Mathematics, Higher Teachers' Training College, University of Yaounde I, PO Box 47, Yaounde, Cameroon 2 Institute of Mathematics, University of Kassel, Heinrich-Plett Street 40, Kassel 34132, Germany 3 Department of Mathematics, Higher Teachers' Training College, University of Maroua, PO Box 55, Maroua, Cameroon
* Author to whom correspondence should be addressed.
Received: 28 May 2013 / Revised: 4 June 2013 / Accepted: 3 July 2013 / Published: 23 July 2013
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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 Askey-Wilson polynomials; nonuniform lattices; difference equations; divided-difference equations; Stieltjes function
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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

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