Caloric polynomial

In differential equations, the "m"th-degree caloric polynomial (or heat polynomial) is a "parabolically "m"-homogeneous" polynomial "P"_"m"("x", "t") that satisfies the heat equation

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

"Parabolically "m"-homogeneous" means

: $P(lambda\; x,\; lambda^2\; t)\; =\; lambda^m\; P(x,t)\; ext\{\; for\; \}lambda\; >\; 0.,$

The polynomial is given by

: $P\_m(x,t)\; =\; sum\_\{ell=0\}^\{lfloor\; m/2\; floor\}\; frac\{m!\}\{ell!(m\; -\; 2ell)!\}\; x^\{m\; -\; 2ell\}\; t^ell.$

It is unique up to a factor.

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

References

*citation|first=John|last=Cannon|title=The One-Dimensional Heat Equation|publisher=Addison-Wesley|year=1984|series=Encyclopedia of mathematics and its applications|isbn=0-521-30243-9

External links

* [http://arxiv.org/pdf/math.AP/0612506.pdf Zeroes of complex caloric functions and singularities of complex viscous Burgers equation]