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

[math]\displaystyle{ \frac{\partial P}{\partial t} = \frac{\partial^2 P}{\partial x^2}. }[/math]

"Parabolically m-homogeneous" means

[math]\displaystyle{ P(\lambda x, \lambda^2 t) = \lambda^m P(x,t)\text{ for }\lambda \gt 0.\, }[/math]

The polynomial is given by

[math]\displaystyle{ P_m(x,t) = \sum_{\ell=0}^{\lfloor m/2 \rfloor} \frac{m!}{\ell!(m - 2\ell)!} x^{m - 2\ell} t^\ell. }[/math]

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