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

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

"Parabolically m-homogeneous" means

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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.

References

• . Contains an extensive bibliography on various topics related to the heat equation.

External links

• Zeroes of complex caloric functions and singularities of complex viscous Burgers equation