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

"Parabolically m-homogeneous" means

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

The polynomial is given by

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

It is unique up to a factor.

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

References

External links

• Zeroes of complex caloric functions and singularities of complex viscous Burgers equation