Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.8 ASSOCIATED LAGUERRE FUNCTIONS

Show that

I≡

∫∞

0

Lmn(x)Lmn(x)xme−xdx=

(n+m)!

n!

. (18.122)

Using the Rodrigues’ formula (18.121), we may write

I=

1

n!

∫∞

0

Lmn(x)

dn

dxn

(xn+me−x)dx=

(−1)n

n!

∫∞

0

dnLmn

dxn

xn+me−xdx,

where, in the second equality, we have integrated by partsntimes and used the fact that
the boundary terms all vanish. From (18.120) we see thatdnLmn/dxn=(−1)n. Thus we have

I=

1

n!

∫∞

0

xn+me−xdx=

(n+m)!

n!

,

where, in the second equality, we use the expression (18.153) defining the gamma function
(see section 18.12).

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

anLmn(x),

in which the coefficientsanare given by

an=

n!

(n+m)!

∫∞

0

f(x)Lmn(x)xme−xdx.

We note that it is sometimes convenient to define theorthogonal associated

Laguerre functionsφmn(x)=xm/^2 e−x/^2 Lmn(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 associated Laguerre polynomials is given by

G(x, h)=

e−xh/(1−h)

(1−h)m+1

=

∑∞

n=0

Lmn(x)hn. (18.123)

This can be obtained by differentiating the generating function (18.114) for the

ordinary Laguerre polynomialsmtimes with respect tox, and using (18.119).

Use the generating function (18.123) to obtain an expression forLmn(0).

From (18.123), we have

∑∞

n=0

Lmn(0)hn=

1

(1−h)m+1

=1+(m+1)h+

(m+1)(m+2)

2!

h^2 +···+

(m+1)(m+2)···(m+n)

n!

hn+···,