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Generalized Segal–Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials

Chadaphorn Kodsueb Orcid Logo, Eugene Lytvynov Orcid Logo

Journal of Mathematical Physics, Volume: 66, Issue: 12

Swansea University Authors: Chadaphorn Kodsueb Orcid Logo, Eugene Lytvynov Orcid Logo

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DOI (Published version): 10.1063/5.0257878

Abstract

Meixner (1934) proved that there exist exactly five classes of orthogonal Sheffer sequences: Hermite polynomials which are orthogonal with respect to Gaussian distribution, Charlier polynomials orthogonal with respect to Poisson distribution, Laguerre polynomials orthogonal with respect to gamma dis...

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Published in: Journal of Mathematical Physics
ISSN: 0022-2488 1089-7658
Published: AIP Publishing 2025
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URI: https://cronfa.swan.ac.uk/Record/cronfa70889
first_indexed 2025-11-13T13:32:03Z
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spelling 2026-01-12T22:18:51.3859685 v2 70889 2025-11-13 Generalized Segal–Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials 65037cc7d8cb5ddc705b10a8b38d3243 0000-0001-7937-6147 Chadaphorn Kodsueb Chadaphorn Kodsueb true false e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2025-11-13 MACS Meixner (1934) proved that there exist exactly five classes of orthogonal Sheffer sequences: Hermite polynomials which are orthogonal with respect to Gaussian distribution, Charlier polynomials orthogonal with respect to Poisson distribution, Laguerre polynomials orthogonal with respect to gamma distribution, Meixner polynomials of the first kind, orthogonal with respect to negative binomial distribution, and Meixner polynomials of the second kind, orthogonal with respect to Meixner distribution. The Segal-Bargmann transform provides a unitary isomorphism between the $L^2$-space of the Gaussian distribution and the Fock or Segal-Bargmann space of entire funcitons. This construction was also extended to the case of the Poisson distribution. The present paper deals with the latter three classes of orthogonal Sheffer sequences. By using a set of nonlinear coherent states, we construct and study a generalized Segal--Bargmann transform which is a unitary isomorphism between the $L^2$-space of the orthogonality measure and a certain Fock space of entire functions. To derive our results, we use normal ordering in generalized Weyl algebras that are naturally associated with the orthogonal Sheffer sequences. Journal Article Journal of Mathematical Physics 66 12 AIP Publishing 0022-2488 1089-7658 3 12 2025 2025-12-03 10.1063/5.0257878 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University Other C.K. was financially supported by the Doctoral Training Program (DTP), EPSRC, UKRI which co-operated with Faculty of Science and Engineering, Swansea University, the project reference 2602423, related to EP/T517987/1. 2026-01-12T22:18:51.3859685 2025-11-13T13:22:57.8306197 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Chadaphorn Kodsueb 0000-0001-7937-6147 1 Eugene Lytvynov 0000-0001-9685-7727 2 70889__35972__97032872295b4eeeb007740420bbfe4a.pdf 70889.VoR.pdf 2026-01-12T22:18:07.4882078 Output 4932571 application/pdf Version of Record true © 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license. true eng https://creativecommons.org/licenses/by/4.0/)
title Generalized Segal–Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials
spellingShingle Generalized Segal–Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials
Chadaphorn Kodsueb
Eugene Lytvynov
title_short Generalized Segal–Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials
title_full Generalized Segal–Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials
title_fullStr Generalized Segal–Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials
title_full_unstemmed Generalized Segal–Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials
title_sort Generalized Segal–Bargmann transforms and generalized Weyl algebras associated with the Meixner class of orthogonal polynomials
author_id_str_mv 65037cc7d8cb5ddc705b10a8b38d3243
e5b4fef159d90a480b1961cef89a17b7
author_id_fullname_str_mv 65037cc7d8cb5ddc705b10a8b38d3243_***_Chadaphorn Kodsueb
e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov
author Chadaphorn Kodsueb
Eugene Lytvynov
author2 Chadaphorn Kodsueb
Eugene Lytvynov
format Journal article
container_title Journal of Mathematical Physics
container_volume 66
container_issue 12
publishDate 2025
institution Swansea University
issn 0022-2488
1089-7658
doi_str_mv 10.1063/5.0257878
publisher AIP Publishing
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
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description Meixner (1934) proved that there exist exactly five classes of orthogonal Sheffer sequences: Hermite polynomials which are orthogonal with respect to Gaussian distribution, Charlier polynomials orthogonal with respect to Poisson distribution, Laguerre polynomials orthogonal with respect to gamma distribution, Meixner polynomials of the first kind, orthogonal with respect to negative binomial distribution, and Meixner polynomials of the second kind, orthogonal with respect to Meixner distribution. The Segal-Bargmann transform provides a unitary isomorphism between the $L^2$-space of the Gaussian distribution and the Fock or Segal-Bargmann space of entire funcitons. This construction was also extended to the case of the Poisson distribution. The present paper deals with the latter three classes of orthogonal Sheffer sequences. By using a set of nonlinear coherent states, we construct and study a generalized Segal--Bargmann transform which is a unitary isomorphism between the $L^2$-space of the orthogonality measure and a certain Fock space of entire functions. To derive our results, we use normal ordering in generalized Weyl algebras that are naturally associated with the orthogonal Sheffer sequences.
published_date 2025-12-03T05:33:52Z
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score 11.096191

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