## Hermite Differential Equation

 (1)

This differential equation has an irregular singularity at . It can be solved using the series method
 (2)

 (3)

Therefore,
 (4)

and
 (5)

for , 2, .... Since (4) is just a special case of (5),
 (6)

for , 1, .... The linearly independent solutions are then

 (7) (8)

If , 4, 8, ..., then terminates with the Power , and (normalized so that the Coefficient of is ) is the regular solution to the equation, known as the Hermite Polynomial. If , 6, 10, ..., then terminates with the Power , and (normalized so that the Coefficient of is ) is the regular solution to the equation, known as the Hermite Polynomial.

If , then Hermite's differential equation becomes

 (9)

which is of the form and so has solution
 (10)