The secondorder 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)
