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

Abstract

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