A note on the discrete Cauchy-Kovalevskaya extension

TL;DR

This paper uses umbral calculus to reformulate the discrete Cauchy-Kovalevskaya extension within hypercomplex variables, providing integral representations and linking to differential-difference Cauchy problems.

Contribution

It introduces a novel umbral calculus approach to the discrete Cauchy-Kovalevskaya extension, incorporating integral representations and connections to differential-difference equations.

Findings

Integral representation as a space-time Fourier inversion formula

Link between the extension and differential-difference Cauchy problems

Use of Laplace transform and Mittag-Leffler functions

Abstract

In this paper we exploit the umbral calculus framework to reformulate the so-called discrete Cauchy-Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for the underlying solution, but also integral representations for the Chebyshev polynomials of first and second kind by means of its Cauchy principal values. It turns out that the resulting integral representation associated to our toy problem is a space-time Fourier type inversion formula. Moreover, with the aid of some Laplace transform identities involving the generalized Mittag-Leffler function we are able to establish a link with a Cauchy problem of differential-difference type.