Hermite polynomials. Hermite’s differential equation. Generating function. Recurrence formulas.

Hermite’s differential equation. The equation

1)        y" - 2xy' + 2ny = 0

If n = 0, 1, 2, 3, .... the solution of Hermite’s equation is given by Hermite’s polynomial H_n(x).

Hermite polynomial. The polynomial given by Rodrigue’s formula

First few Hermite polynomials

Generating function

Recurrence formulas

Orthogonality of Hermite polynomials

Orthogonal series

where

Miscellaneous formulas

                                          Source: Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)

References.

1. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)