Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions
Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its q-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite suppo...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209778 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions / M.E.H. Ismail, E. Koelink, P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862714893041401856 |
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| author | Ismail, M.E.H. Koelink, E. Román, P. |
| author_facet | Ismail, M.E.H. Koelink, E. Román, P. |
| citation_txt | Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions / M.E.H. Ismail, E. Koelink, P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its q-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give an alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek, and big q-Jacobi polynomials and big q-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek, and Krawtchouk polynomials.
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| first_indexed | 2025-12-07T17:54:28Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209778 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:54:28Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
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| spelling | Ismail, M.E.H. Koelink, E. Román, P. 2025-11-26T12:18:22Z 2018 Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions / M.E.H. Ismail, E. Koelink, P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C45; 33D45; 42C05; 37K10 arXiv: 1802.09190 https://nasplib.isofts.kiev.ua/handle/123456789/209778 https://doi.org/10.3842/SIGMA.2018.072 Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its q-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give an alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek, and big q-Jacobi polynomials and big q-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek, and Krawtchouk polynomials. Mourad Ismail acknowledges the research support and hospitality of Radboud University during his visits, which initiated this collaboration. Erik Koelink gratefully acknowledges the support of FaMAF as an invited professor at Universidad Nacional de Córdoba for a research visit. The work of Pablo Román was supported by Radboud Excellence Fellowship, CONICET grant PIP 112-200801-01533, FONCyT grant PICT 2014-3452, and by SeCyT-UNC. We thank a referee for useful advice. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions Article published earlier |
| spellingShingle | Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions Ismail, M.E.H. Koelink, E. Román, P. |
| title | Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions |
| title_full | Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions |
| title_fullStr | Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions |
| title_full_unstemmed | Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions |
| title_short | Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions |
| title_sort | generalized burchnall-type identities for orthogonal polynomials and expansions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209778 |
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