On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2013-07-01
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| Series: | Axioms |
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| Online Access: | http://www.mdpi.com/2075-1680/2/3/404 |
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| author | Salifou Mboutngam Maurice Kenfack-Nangho Mama Foupouagnigni Wolfram Koepf |
| author_facet | Salifou Mboutngam Maurice Kenfack-Nangho Mama Foupouagnigni Wolfram Koepf |
| author_sort | Salifou Mboutngam |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-12T19:05:39Z |
| format | Article |
| id | doaj.art-ee2d630695304874aceba7531af0ce1e |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-04-12T19:05:39Z |
| publishDate | 2013-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-ee2d630695304874aceba7531af0ce1e2022-12-22T03:20:01ZengMDPI AGAxioms2075-16802013-07-012340443410.3390/axioms2030404On Solutions of Holonomic Divided-Difference Equations on Nonuniform LatticesSalifou MboutngamMaurice Kenfack-NanghoMama FoupouagnigniWolfram KoepfThe 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.http://www.mdpi.com/2075-1680/2/3/404Askey-Wilson polynomialsnonuniform latticesdifference equationsdivided-difference equationsStieltjes function |
| spellingShingle | Salifou Mboutngam Maurice Kenfack-Nangho Mama Foupouagnigni Wolfram Koepf On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices Axioms Askey-Wilson polynomials nonuniform lattices difference equations divided-difference equations Stieltjes function |
| title | On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices |
| title_full | On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices |
| title_fullStr | On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices |
| title_full_unstemmed | On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices |
| title_short | On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices |
| title_sort | on solutions of holonomic divided difference equations on nonuniform lattices |
| topic | Askey-Wilson polynomials nonuniform lattices difference equations divided-difference equations Stieltjes function |
| url | http://www.mdpi.com/2075-1680/2/3/404 |
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