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

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

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, https://books.google.com/books?id=XWSnBZxbz2oC . 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