Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

The above orthogonality and normalisation conditions allow us to expand any

(reasonable) function in the interval 0≤x<∞in a series of the form

f(x)=

∑∞

n=0

anLn(x),

in which the coefficientsanare given by

an=

∫∞

0

f(x)Ln(x)e−xdx.

We note that it is sometimes convenient to define theorthonormal Laguerre func-

tionsφn(x)=e−x/^2 Ln(x), which may also be used to produce a series expansion

of a function in the interval 0≤x<∞.

Generating function

The generating function for the Laguerre polynomials is given by

G(x, h)=

e−xh/(1−h)

1 −h

=

∑∞

n=0

Ln(x)hn. (18.114)

We may prove this result by differentiating the generating function with respect to

xandh, respectively, to obtain recurrence relations for the Laguerre polynomials,

which may then be combined to show that the functionsLn(x) in (18.114) do

indeed satisfy Laguerre’s equation (as discussed in the next subsection).

Recurrence relations

The Laguerre polynomials obey a number of useful recurrence relations. The

three most important relations are as follows:

(n+1)Ln+1(x)=(2n+1−x)Ln(x)−nLn− 1 (x), (18.115)

Ln− 1 (x)=L′n− 1 (x)−L′n(x), (18.116)

xL′n(x)=nLn(x)−nLn− 1 (x). (18.117)

The first two relations are easily derived from the generating function (18.114),

and may be combined straightforwardly to yield the third result.

Derive the recurrence relations (18.115) and (18.116).

Differentiating the generating function (18.114) with respect toh, we find

∂G

∂h

=

(1−x−h)e−xh/(1−h)

(1−h)^3

=

∑

nLnhn−^1.

Thus, we may write

(1−x−h)

∑

Lnhn=(1−h)^2

∑

nLnhn−^1 ,

and, on equating coefficients ofhnon each side, we obtain

(1−x)Ln−Ln− 1 =(n+1)Ln+1− 2 nLn+(n−1)Ln− 1 ,