Discrete Fourier-Jacobi transform

TL;DR

This paper introduces and studies discrete Fourier-Jacobi transforms involving hypergeometric functions, providing inversion formulas and analyzing their properties for suitable functions and sequences.

Contribution

It presents the first discrete analogs of the classical Fourier-Jacobi transform, including new inversion formulas and analysis of their properties.

Findings

Established inversion formulas for the discrete Fourier-Jacobi transform.

Analyzed the properties of series and integrals involving hypergeometric functions.

Provided conditions for the applicability of these transforms.

Abstract

Discrete analogs of the classical Fourier-Jacobi transform are introduced and investigated. It involves series and integrals with respect to parameters of the Gauss hypergeometric function ${}_2F_1(a+in/2,a-in/2;\ c; -x^2 ), \ x >0, n \in \mathbb{N}, a,c > 0, i $ is the imaginary unit. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established.