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

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

"Parabolically m-homogeneous" means

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

The polynomial is given by

P m ( x , t ) = ∑ ℓ = 0 ⌊ m / 2 ⌋ m ! ℓ ! ( m − 2 ℓ ) ! x m − 2 ℓ t ℓ . {\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 = −1/2, 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.
  • Zeroes of complex caloric functions and singularities of complex viscous Burgers equation


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