Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.8 ASSOCIATED LAGUERRE FUNCTIONS

which trivially rearranges to give the recurrence relation (18.115).
To obtain the recurrence relation (18.116), webegin by differentiating the generating
function (18.114) with respect tox, which yields

∂G

∂x

=−

he−xh/(1−h)

(1−h)^2

=

∑

L′nhn,

and thus we have

−h

∑

Lnhn=(1−h)

∑

L′nhn.

Equating coefficients ofhnon each side then gives

−Ln− 1 =L′n−L′n− 1 ,

which immediately simplifies to give (18.116).

18.8 Associated Laguerre functions

The associated Laguerre equation has the form

xy′′+(m+1−x)y′+ny= 0; (18.118)

it has a regular singularity atx= 0 and an essential singularity atx=∞.We

restrict our attention to the situation in which the parametersnandmare both

non-negative integers, as is the case in nearly all physical problems. The associated

Laguerre equation occurs most frequently in quantum-mechanical applications.

Any solution of (18.118) is called anassociated Laguerre function.

Solutions of (18.118) for non-negative integers nandmare given by the

associated Laguerre polynomials

Lmn(x)=(−1)m

dm

dxm

Ln+m(x), (18.119)

whereLn(x) are the ordinary Laguerre polynomials.§

Show that the functionsLmn(x)defined in (18.119) are solutions of (18.118).

Since the Laguerre polynomialsLn(x) are solutions of Laguerre’s equation (18.107), we
have

xL′′n+m+(1−x)L′n+m+(n+m)Ln+m=0.

Differentiating this equationmtimes using Leibnitz’ theorem and rearranging, we find

xLn(m++2)m+(m+1−x)L(nm++1)m+nL(nm+)m=0.

On multiplying through by (−1)mand settingLmn=(−1)mL(nm+)m, in accord with (18.119),
we obtain

x(Lmn)′′+(m+1−x)(Lmn)′+nLmn=0,

which shows that the functionsLmnare indeed solutions of (18.118).

§Note that some authors define the associated Laguerre polynomials asLmn(x)=(dm/dxm)Ln(x),

which is thus related to our expression (18.119) byLmn(x)=(−1)mLmn+m(x).