Laguerre Differential Equation

The Laguerre differential equation is given by

(1)

Equation (1) is a special case of the more general associated Laguerre differential equation, defined by

(2)

where and are real numbers (Iyanaga and Kawada 1980, p. 1481; Zwillinger 1997, p. 124) with .

The general solution to the associated equation (2) is

(3)

Note that in the special case , the associated Laguerre differential equation is of the form

(4)

so the solution can be found using an integrating factor

where is the En-function.

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,

	(12)
	(13)
	(14)
$[(nu+1)a_1+lambdaa_0]+sum_(n=1)^(infty){[(n+1)n+(nu+1)(n+1)]a_(n+1)-na_n+lambdaa_n}x^n=0$	(15)
	(16)

This requires

			(17)
			(18)

for . Therefore,

(19)

for , 2, ..., so

If is a nonnegative integer, then the series terminates and the solution is given by

(23)

where is an associated Laguerre polynomial and is a Pochhammer symbol. In the special case , the associated Laguerre polynomial collapses to a usual Laguerre polynomial and the solution collapses to

(24)

See also