Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

Generating function

The generating function for associated Legendre functions can be easily derived

by combining their definition (18.32) with the generating function for the Legendre

polynomials given in (18.15). We find that

G(x, h)=

(2m)!(1−x^2 )m/^2

2 mm!(1− 2 hx+h^2 )m+1/^2

=

∑∞

n=0

Pnm+m(x)hn. (18.40)

Derive the expression (18.40) for the associated Legendre generating function.

The generating function (18.15) for the Legendre polynomials reads

∑∞

n=0

Pnhn=(1− 2 xh+h^2 )−^1 /^2.

Differentiating both sides of this resultmtimes (assumimgmto be non-negative), mutli-
plying through by (1−x^2 )m/^2 and using the definition (18.32) of the associated Legendre
functions, we obtain

∑∞

n=0

Pnmhn=(1−x^2 )m/^2

dm

dxm

(1− 2 xh+h^2 )−^1 /^2.

Performing the derivatives on the RHS gives

∑∞

n=0

Pnmhn=

1 · 3 · 5 ···(2m−1)(1−x^2 )m/^2 hm

(1− 2 xh+h^2 )m+1/^2

.

Dividing through byhm, re-indexing the summation on the LHS and noting that, quite
generally,

1 · 3 · 5 ···(2r−1) =

1 · 2 · 3 ··· 2 r

2 · 4 · 6 ··· 2 r

=

(2r)!

2 rr!

,

we obtain the final result (18.40).

Recurrence relations

As one might expect, the associated Legendre functions satisfy certain recurrence

relations. Indeed, the presence of the two indicesnandmmeans that a much wider

range of recurrence relations may be derived. Here we shall content ourselves

with quoting just four of the most useful relations:

Pnm+1=

2 mx

(1−x^2 )^1 /^2

Pnm+[m(m−1)−n(n+1)]Pnm−^1 , (18.41)

(2n+1)xPnm=(n+m)Pnm− 1 +(n−m+1)Pnm+1, (18.42)

(2n+ 1)(1−x^2 )^1 /^2 Pnm=Pnm+1+1−Pnm−+1 1 , (18.43)

2(1−x^2 )^1 /^2 (Pnm)′=Pnm+1−(n+m)(n−m+1)Pnm−^1. (18.44)

We note that, by virtue of our adopted definition (18.32), these recurrence relations

are equally valid for negative and non-negative values ofm. These relations may