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). The ordinary Laguerre differential equation has .

For , the general solution to the associated equation (2) is

(3)

where is a confluent hypergeometric function of the second kind. The value at is taken on the upper side of its branch cut (NIST, §13.2(v)). The functions and the generalized Laguerre polynomial are not always independent. For nonnegative integer ,

(4)

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

(5)

so the solution can be found using an integrating factor

where is the E_n-function.

The associated Laguerre differential equation has a regular singular point at 0 and an irregular singularity at . For , , ..., a solution analytic at 0 can be obtained using a series expansion,

	(13)
	(14)
	(15)
$[(nu+1)a_1+lambdaa_0]+sum_(n=1)^(infty){[(n+1)n+(nu+1)(n+1)]a_(n+1)-na_n+lambdaa_n}x^n=0$	(16)
	(17)

This requires

			(18)
			(19)

for . Thus

Under this condition on , if is a nonnegative integer, then the series terminates and the analytic 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)

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1481, 1980.

NIST. "Kummer Functions." Digital Library of Mathematical Functions, §§13.2 and 13.6. https://dlmf.nist.gov/13.2, https://dlmf.nist.gov/13.6.

See also