Girish Mahajan (Editor)

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

∂ P ∂ t = ∂ 2 P ∂ x 2 .

"Parabolically m-homogeneous" means

P ( λ x , λ 2 t ) = λ m P ( x , t )  for  λ > 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 .

It is unique up to a factor.

With t = −1, this polynomial reduces to the mth-degree Hermite polynomial in x.

References

Caloric polynomial Wikipedia
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