Polynomial Expansions for Solutions of Higher-Order q-Bessel Heat Equation

TL;DR

This paper introduces a q-analogue of higher-order Bessel operators, establishing their properties and developing a q-Fourier transform and heat polynomial theory, extending classical results to the q-calculus setting.

Contribution

It constructs a new q-analogue of higher-order Bessel operators, explores their properties, and develops associated Fourier and heat polynomial theories.

Findings

Established properties of the q-analogue Bessel functions.

Constructed the q-Fourier transform for these operators.

Extended classical heat polynomial theory to the q-calculus context.

Abstract

In this paper we give the q-analogue of the higher-order Bessel operators studied by M. I. Klyuchantsev [12] and A. Fitouhi, N. H. Mahmoud and S. A. Ould Ahmed Mahmoud [3]. Our objective is twofold. First, using the q-Jackson integral and the q-derivative, we aim at establishing some properties of this function with proofs similar to the classical case. Second our goal is to construct the associated q-Fourier transform and the q-analogue of the theory of the heat polynomials introduced by P. C. Rosenbloom and D. V. Widder [13]. Our operator for some value of the vector index generalize the q-j_\alpha Bessel operator of the second order in [4] and a q-Third operator in [6].