Airy-heat functions, Hermite and higher order Hermite generating functions

TL;DR

This paper explores the connections between Hermite polynomials, Airy-Heat functions, and higher order PDEs, demonstrating their equivalence under certain conditions and enriching the understanding of their interrelations.

Contribution

It establishes the equivalence between generating functions of Hermite polynomials, generalized Airy-Heat equations, and higher order PDEs, providing new insights into their relationships.

Findings

Demonstrated the equivalence of different mathematical formulations.

Connected Hermite polynomial generating functions with Airy-Heat functions.

Extended the understanding of higher order PDEs and their solutions.

Abstract

In this note we discuss the relationship between the generating functions of some Hermite polynomials $H$, $ \sum\limits_{j=0}^\infty H_{j\cdot n}(u) z^n/n!$, generalized Airy-Heat equations $(1/2\pi)\int_{-\infty}^{+\infty}\exp\{a(i\lambda)^n-(1/2)\lambda^2t+i\lambda x\}d\lambda$, higher order PDE's $(\partial u/\partial t)(t,x)=a(\partial^nu/\partial x^n)(t,x)+(1/2)s(\partial^2u/\partial x^2)(t,x)$, and generating functions of higher order Hermite polynomials $H^{(n)}$: $\sum\limits_{j=0}^\infty H^{(n)}_j(v)x^j/j!$. In particular, we show that under some conditions, these problems are equivalent.