Preprint
On Solutions of Holonomic Divided-Difference Equations on Non-Uniform Lattices
Zusammenfassung
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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@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},
year={2010}
}
0500 Oax 0501 Text $btxt$2rdacontent 0502 Computermedien $bc$2rdacarrier 1100 2010$n2010 1500 1/eng 2050 ##0##urn:nbn:de:hebis:34-2010082534270 3000 Foupouagnigni, Mama 3010 Koepf, Wolfram 3010 Kenfack Nangho, Maurice 3010 Mboutngam, Salifou 4000 On Solutions of Holonomic Divided-Difference Equations on Non-Uniform Lattices / Foupouagnigni, Mama 4030 4060 Online-Ressource 4085 ##0##=u http://nbn-resolving.de/urn:nbn:de:hebis:34-2010082534270=x R 4204 \$dPreprint 4170 Mathematische Schriften Kassel ;; 10, 03 7136 ##0##urn:nbn:de:hebis:34-2010082534270
2010-08-25T07:53:49Z 2010-08-25T07:53:49Z 2010 urn:nbn:de:hebis:34-2010082534270 http://hdl.handle.net/123456789/2010082534270 eng Urheberrechtlich geschützt https://rightsstatements.org/page/InC/1.0/ Askey-Wilson polynomials Non-uniform lattices Difference equations Divided-difference equations Stieltjes function 510 On Solutions of Holonomic Divided-Difference Equations on Non-Uniform Lattices Preprint 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]. open access Foupouagnigni, Mama Koepf, Wolfram Kenfack Nangho, Maurice Mboutngam, Salifou Mathematische Schriften Kassel ;; 10, 03 33D45 39A13 Mathematische Schriften Kassel 10, 03
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