Hellenica World

# .

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

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

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.
References

Cannon, John Rozier (1984), The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications 23 (1st ed.), Reading/Cambridge: Addison-Wesley Publishing Company/Cambridge University Press, pp. XXV+483, ISBN 978-0-521-30243-2, MR 0747979, Zbl 0567.35001. Contains an extensive bibliography on various topics related to the heat equation.