Hermite Differential Equation

The second-order ordinary 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)

These can be done in closed form as

			(9)
			(10)

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

			(11)
			(12)
			(13)

where  is the erfi function.

If , then Hermite's differential equation becomes

(14)

which is of the form  and so has solution

			(15)
			(16)
			(17)