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Meixner $d$-Orthogonal Polynomials Arising from $\mathfrak{su}(1,1)$

Borhen Halouani, Fethi Bouzeffour

Abstract

In this study, we present a novel family of Meixner-type $d$-orthogonal polynomials, which are distinguished as a particular subset of multiple orthogonal polynomials. We demonstrate their connection to the Lie algebra $\mathfrak{su}(1,1)$ by identifying them as matrix elements of an appropriately defined nonlinear operator. Utilizing Barut-Girardello coherent states, we explicitly outline their key features, including recurrence relations, generating functions, and $d$-orthogonality relations, among others.

Meixner $d$-Orthogonal Polynomials Arising from $\mathfrak{su}(1,1)$

Abstract

In this study, we present a novel family of Meixner-type -orthogonal polynomials, which are distinguished as a particular subset of multiple orthogonal polynomials. We demonstrate their connection to the Lie algebra by identifying them as matrix elements of an appropriately defined nonlinear operator. Utilizing Barut-Girardello coherent states, we explicitly outline their key features, including recurrence relations, generating functions, and -orthogonality relations, among others.

Paper Structure

This paper contains 13 sections, 5 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. $d$-Orthogonal Polynomials
  3. Non linear coherent states.
  4. $\mathfrak{su}(1,1)$ Lie algebra.
  5. Some identities in $\mathfrak{su}(1,1)$
  6. Matrix Elements of an Operator and $d$-Orthogonal Polynomials
  7. Recurrence Relation
  8. Generating function
  9. Connection with the First Kind Meixner Orthogonal Polynomials
  10. Lowering operator
  11. $d$-orthogonal polynomials of Meixner type
  12. Recurrence relation of $\phi_{nk}$ and linear functional vector
  13. Explicit expression of ${\bf{\phi}}_{ik}$

Key Result

Proposition \oldthetheorem

The matrix elements $\psi_{nk}$ are expressed as where $\{P_n(k)\}_{n\geq0}$ is a $d$-OPS in the argument $k$, satisfying the recurrence relation of order $d+1$ with the initial conditions $P_0(k) = 1$ and $P_n(k) = 0$ for $n < 0$.

Theorems & Definitions (5)

  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem