Caloric polynomial

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


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

"Parabolically m-homogeneous" means


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

The polynomial is given by


 * $$ P_m(x,t) = \sum_{\ell=0}^{\lfloor m/2 \rfloor} \frac{m!}{\ell!(m - 2\ell)!} x^{m - 2\ell} t^\ell. $$

It is unique up to a factor.

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