Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.7 LAGUERRE FUNCTIONS

Prove that the expression (18.112) yields thenth Laguerre polynomial.

Evaluating thenth derivative in (18.112) using Leibnitz’ theorem, we find

Ln(x)=

ex

n!

∑n

r=0

nCrd

rxn

dxr

dn−re−x

dxn−r

=

ex

n!

∑n

r=0

n!

r!(n−r)!

n!

(n−r)!

xn−r(−1)n−re−x

=

∑n

r=0

(−1)n−r

n!

r!(n−r)!(n−r)!

xn−r.

Relabelling the summation using the indexm=n−r,weobtain

Ln(x)=

∑n

m=0

(−1)m

n!

(m!)^2 (n−m)!

xm,

which is precisely the expression (18.111) for thenth Laguerre polynomial.

Mutual orthogonality

In section 17.4, we noted that Laguerre’s equation could be put into Sturm–

Liouville form withp=xe−x,q=0,λ=νandρ=e−x, and its natural interval

is thus [0,∞]. Since the Laguerre polynomialsLn(x)aresolutionsoftheequation

and are regular at the end-points, they must be mutually orthogonal over this

interval with respect to the weight functionρ=e−x,i.e.
∫∞

0

Ln(x)Lk(x)e−xdx=0 ifn=k.

This result may also be proved directly using the Rodrigues’ formula (18.112).

Indeed, the normalisation, whenk=n, is most easily found using this method.

Show that

I≡

∫∞

0

Ln(x)Ln(x)e−xdx=1. (18.113)

Using the Rodrigues’ formula (18.112), we may write

I=

1

n!

∫∞

0

Ln(x)

dn

dxn

(xne−x)dx=

(−1)n

n!

∫∞

0

dnLn

dxn

xne−xdx,

where, in the second equality, we have integrated by partsntimes and used the fact that the
boundary terms all vanish. WhendnLn/dxnis evaluated using (18.111), only the derivative
of them=nterm survives and that has the value [ (−1)nn!n!]/[(n!)^2 0!] = (−1)n. Thus
we have

I=

1

n!

∫∞

0

xne−xdx=1,

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