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On Solutions of Holonomic Divided-Difference Equations on Non-Uniform Lattices

The main aim of this paper is the development of suitable bases (replacing the power basis x^n (n\in\IN_\le 0) which enable the direct series representation of orthogonal polynomial systems on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows to write solutions of arbitrary divided-difference equations in terms of series representations extending results given in [16] for the q-case. Furthermore it enables the representation of the Stieltjes function which can be 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 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 [5], see also [6].

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Citation
In: Mathematische Schriften Kassel 10, 03 / (2010) , S. ;
@article{urn:nbn:de:hebis:34-2010082534270,
  author    ={Foupouagnigni, Mama and Koepf, Wolfram and Kenfack Nangho, Maurice and Mboutngam, Salifou},
  title    ={On Solutions of Holonomic Divided-Difference Equations on Non-Uniform Lattices},
  copyright  ={https://rightsstatements.org/page/InC/1.0/},
  language ={en},
  year   ={2010}
}