Caloric polynomial

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In differential equations, the mth-degree caloric polynomial (or heat polynomial) is a "parabolically m-homogeneous" polynomial Pm(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.

[edit] References

  • Cannon, John (1984), The One-Dimensional Heat Equation, Encyclopedia of mathematics and its applications, Addison-Wesley, ISBN 0-521-30243-9 

[edit] External links