Discrete Mehler-Fock transforms

TL;DR

This paper introduces and studies discrete versions of the classical Mehler-Fock transforms, involving series with associated Legendre functions and incomplete Legendre integrals, and establishes various function expansions.

Contribution

It presents new discrete analogs of the Mehler-Fock transforms and derives expansion formulas for functions and sequences using these series and integrals.

Findings

Established several expansions of functions and sequences.

Analyzed properties of the introduced discrete transforms.

Connected the transforms with incomplete Legendre integrals.

Abstract

Discrete analogs of the classical Mehler-Fock transforms are introduced and investigated. It involves series with the associated Legendre function $P^\mu_{in-1/2}(x), x > 1,\ {\rm Re} \mu < 1/2, \ n \in \mathbb{N}, i $ is the imaginary unit, and the so-called incomplete Legendre integrals. Several expansions of arbitrary functions and sequences in terms of these series and integrals are established.