Caloric polynomial

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In differential equations, the mth-degree caloric polynomial (or heat polynomial) is a "parabolically m-homogeneous" polynomial Pm(xt) 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 = −1, this polynomial reduces to the mth-degree Hermite polynomial in x.


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