Hermite Differential Equation

The second-order ordinary differential equation

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

Therefore,

and

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

for , 1, ....

The linearly independent solutions are then

These can be done in closed form as

where is a confluent hypergeometric function of the first kind and is a Hermite polynomial. In particular, for , 2, 4, ..., the solutions can be written

where is the erfi function.

If , then Hermite's differential equation becomes

which is of the form and so has solution