Caloric polynomial

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

"Parabolically m-homogeneous" means

The polynomial is given by

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, vol. 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.


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