Caloric polynomial

In differential equations, the mth-degree caloric polynomial (or heat polynomial) is a "parabolically m-homogeneous" polynomial P_m(x, t) that satisfies the heat equation

${\displaystyle {\frac {\partial P}{\partial t}}={\frac {\partial ^{2}P}{\partial x^{2}}}.}$

"Parabolically m-homogeneous" means

${\displaystyle P(\lambda x,\lambda ^{2}t)=\lambda ^{m}P(x,t){\text{ for }}\lambda >0.\,}$

The polynomial is given by

${\displaystyle P_{m}(x,t)=\sum _{\ell =0}^{\lfloor m/2\rfloor }{\frac {m!}{\ell$ !(m-2\ell )!}}x^{m-2\ell }t^{\ell }.}

It is unique up to a factor.

With t = −1, this polynomial reduces to the mth-degree Hermite polynomial in x.