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In this note we discuss the relationship between the generating functions ofsome 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\lambdax\}d\lambda$, higher order PDE's $(\partial u/\partialt)(t,x)=a(\partial^nu/\partial x^n)(t,x)+(1/2)s(\partial^2u/\partialx^2)(t,x)$, and generating functions of higher order Hermite polynomials$H^{(n)}$: $\sum\limits_{j=0}^\infty H^{(n)}_j(v)x^j/j!$. In particular, weshow that under some conditions, these problems are equivalent.

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