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

Chadaphorn Kodsueb, Eugene Lytvynov

TL;DR

The paper develops a generalized Segal–Bargmann framework for the Meixner class of orthogonal Sheffer sequences by constructing nonlinear coherent states and a unitary transform between $L^2$ spaces tied to Meixner-type measures and a Fock-type space ${\mathbb F}_{\eta,\sigma}(\mathbb C)$. It provides explicit integral representations of the coherent states and shows how multiplication by the variable corresponds to a generalized Weyl algebra, enabling a unified treatment of gamma, negative-binomial, and Meixner distributions. The approach relies on normal ordering in generalized Weyl algebras, embedding results for ${\mathcal E}_{\mathrm{min}}^1(\mathbb C)$, and detailed proofs in sections 4.1 and 4.1.2, culminating in robust integral transforms and operator correspondences. This framework broadens SB-transform theory beyond Gaussian/Poisson to Meixner-type families, with potential extensions to infinite dimensions.

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 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.

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

TL;DR

The paper develops a generalized Segal–Bargmann framework for the Meixner class of orthogonal Sheffer sequences by constructing nonlinear coherent states and a unitary transform between spaces tied to Meixner-type measures and a Fock-type space . It provides explicit integral representations of the coherent states and shows how multiplication by the variable corresponds to a generalized Weyl algebra, enabling a unified treatment of gamma, negative-binomial, and Meixner distributions. The approach relies on normal ordering in generalized Weyl algebras, embedding results for , and detailed proofs in sections 4.1 and 4.1.2, culminating in robust integral transforms and operator correspondences. This framework broadens SB-transform theory beyond Gaussian/Poisson to Meixner-type families, with potential extensions to infinite dimensions.

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 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 -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 -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.
Paper Structure (8 sections, 24 theorems, 152 equations)

This paper contains 8 sections, 24 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. The spaces ${\mathbb F}_{ \eta, \sigma} (\mathbb C)$ and $\mathcal{E}_{\mathrm{min}}^1(\mathbb C)$
  3. Main results
  4. Proofs
  5. Proof of Lemmas \ref{['yufd6e64w5']}--\ref{['xdesrea5qw3']}
  6. The case of the gamma distribution and the negative binomial distribution
  7. The case of the Meixner distribution
  8. The remaining proofs

Key Result

Proposition 2.1

Let $\eta>0$ and $\sigma>0$. Then ${\mathbb F}_{ \eta, \sigma} (\mathbb C)$ is the closed subspace of $L^2 (\mathbb C, \lambda_{\eta, \sigma})$ constructed as the closure of $\mathcal{P}( \mathbb C)$. Furthermore,

Theorems & Definitions (44)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5: Grabiner
  • Corollary 2.6
  • proof
  • Theorem 3.1
  • ...and 34 more