Computation of Fourier transform representations involving the generalized Bessel matrix polynomials

M Abdalla,M Akel

doi:10.1186/s13662-021-03572-w

Abstract

Motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels, in this article we derive the formulas for Fourier cosine and sine transforms of matrix functions involving generalized Bessel matrix polynomials. With the help of these transforms several results are obtained, which are extensions of the corresponding results in the standard cases. The results given here are of general character and can yield a number of (known and new) results in modern integral transforms.

Highlights

In the past few decades, the orthogonal matrix polynomials have attracted a lot of research interest due to their close relations and various applications in many areas of mathematics, engineering, probability theory, graph theory, and physics; for example, see [1,2,3,4,5,6,7,8,9]
Motivated by some of these aforementioned investigations of the Fourier transforms of matrix-valued orthogonal polynomials, in our investigation here we study the Fouriertype transforms of the generalized Bessel matrix polynomials Yn(ξ ; F, L), ξ ∈ C, for matrix parameters F and L
3 Statement and proof of main theorems we investigate several new interesting Fourier cosine and sine transforms of functions involving generalized Bessel matrix polynomials asserted in the following theorems: Theorem 3.1 Let S, F and L be commuting matrices in Cn×n, and let Yn(λξ ; F, L) be given in (15)

Summary

Introduction

In the past few decades, the orthogonal matrix polynomials have attracted a lot of research interest due to their close relations and various applications in many areas of mathematics, engineering, probability theory, graph theory, and physics; for example, see [1,2,3,4,5,6,7,8,9]. Groenevelt and Koelink [42] discussed the generalized Fourier transform with hypergeometric function and matrix-valued orthogonal polynomials as kernels. Definition 2.3 ([4, 45]) Let F be a positive stable matrix in Cn×n.

Results

Conclusion