
(1) 
The Laguerre differential equation is a special case of the more general ``associated Laguerre differential equation''

(2) 
with . Note that if , then the solution to the associated Laguerre differential equation is of the
form

(3) 
and the solution can be found using an Integrating Factor
so

(5) 
The associated Laguerre differential equation has a Regular Singular Point at 0 and an Irregular
Singularity at . It can be solved using a series expansion,



(6) 



(7) 



(8) 

(9) 

(10) 
This requires
for . Therefore,

(13) 
for , 2, ..., so

(14) 
If is a Positive Integer, then the series terminates and the solution is a Polynomial, known
as an associated Laguerre Polynomial (or, if , simply a
Laguerre Polynomial).
See also Laguerre Polynomial
© 19969 Eric W. Weisstein
19990526