Theory of relativistic heat polynomials and one-sided L\'evy distributions

TL;DR

This paper explores the connections between fractional diffusion, relativistic quantum mechanics, and special polynomial families, introducing generalized relativistic heat polynomials for solving relevant differential equations.

Contribution

It uncovers new links between fractional diffusion, relativistic quantum equations, and Carlitz-related polynomials, introducing generalized relativistic heat polynomials for practical problem-solving.

Findings

Identified links between fractional diffusion and relativistic quantum equations.

Introduced generalized relativistic heat polynomials.

Demonstrated applications in solving practical differential equations.

Abstract

The theory of pseudo-differential operators is a powerful tool to deal with differential equations involving differential operators under the square root sign. These type of equations are pivotal elements to treat problems in anomalous diffusion and in relativistic quantum mechanics. In this paper we report on new and unsuspected links between fractional diffusion, quantum relativistic equations and particular families of polynomials, linked to the Carlitz family, and playing the role of relativistic heat polynomials. We introduce generalizations of these polynomial families and point out their specific use for the solutions of problems of practical importance.