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

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

"Parabolically m-homogeneous" means

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

The polynomial is given by

${\displaystyle P_{m}(x,t)=\sum _{\ell =0}^{\lfloor m/2\rfloor }{\frac {m!}{\ell$ !(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.

External links

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