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On q-Bessel matrix polynomials

Ayman Shehata, M. Tawfik, Ayman M. Mahmoud, Nada Mostafa

TL;DR

The paper develops a matrix valued q-analogue of Bessel polynomials (q-BMPs) by defining J_{n,q}(z;A) through a basic hypergeometric matrix function and deriving a comprehensive set of analytic properties. It establishes a second-order q-differential equation, a hierarchy of q-differential and derivative relations, a pure matrix q-recurrence, and transform representations (q-Laplace and q-Mellin), alongside a product formula and integral representations. Furthermore, it reveals connections between q-BMPs and q-Horn's matrix functions H6 and Phi1, enriching the theory of matrix valued basic hypergeometric functions. The framework provides a versatile toolkit for applications in mathematical physics and operator theory where q-deformed, matrix-valued special functions arise.

Abstract

The aim of the present study is to establish some properties for q-Bessel matrix polynomials such as several q-differential matrix equation, q-differential matrix relations and q-recurrence matrix relations, and integral representation, q-Laplace and q-Mellin transforms with the help of q-Analysis. Furthermore, we give connections between q-Horn's matrix functions of two variables and q-Bessel matrix polynomials are given.

On q-Bessel matrix polynomials

TL;DR

The paper develops a matrix valued q-analogue of Bessel polynomials (q-BMPs) by defining J_{n,q}(z;A) through a basic hypergeometric matrix function and deriving a comprehensive set of analytic properties. It establishes a second-order q-differential equation, a hierarchy of q-differential and derivative relations, a pure matrix q-recurrence, and transform representations (q-Laplace and q-Mellin), alongside a product formula and integral representations. Furthermore, it reveals connections between q-BMPs and q-Horn's matrix functions H6 and Phi1, enriching the theory of matrix valued basic hypergeometric functions. The framework provides a versatile toolkit for applications in mathematical physics and operator theory where q-deformed, matrix-valued special functions arise.

Abstract

The aim of the present study is to establish some properties for q-Bessel matrix polynomials such as several q-differential matrix equation, q-differential matrix relations and q-recurrence matrix relations, and integral representation, q-Laplace and q-Mellin transforms with the help of q-Analysis. Furthermore, we give connections between q-Horn's matrix functions of two variables and q-Bessel matrix polynomials are given.
Paper Structure (4 sections, 19 theorems, 91 equations)

This paper contains 4 sections, 19 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derivation of main results
  4. Concluding remarks

Key Result

Theorem 1.1

If $\Omega(\xi)$ and $\Phi(\xi)$ are holomorphic functions of complex variable $\xi$, which are defined in an open set $\Theta$ of complex plane, then (see du) where $\mathbf{F}$, $\mathbf{E}$ are commutative matrices in $\Bbb{C}^{\ell\times \ell}$ with $\sigma(\mathbf{E}) \subset \Theta$ and $\sigma(\mathbf{F})\subset\Theta$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Lemma 1.1
  • Definition 1.8
  • ...and 38 more