þ l À m
ð
Þ l þ m þ 1
ð
Þ C
mþ
1
2
lÀm x
ð Þ ¼ 0:
ð3:178Þ
Once again comparing (3.174) and (3.178), we obtain
d
m P l x
ð Þ
dx
m
¼ constant Á C
mþ
1
2
lÀm x
ð Þ 0 m l
ð
Þ :
ð3:179Þ
Next, let us determine the constant appearing in (3.179). To this end, we consider
a following generating function of the polynomials C
λ
n x
ð Þ defined by [7, 8]
1 À 2tx þ t
2
À
Á Àλ
X 1
n¼0
C
λ
n x
ð Þt
n
λ > À
1
2
:
ð3:180Þ
To calculate (3.180), let us think of a following expression for x and λ:
1 þ x
ð
Þ
Àλ ¼
X 1
m¼0
Àλ
m
x
m ,
ð3:181Þ
where λ is an arbitrary real number and we define
Àλ
m
as
Àλ
m
Àλ Àλ À 1
ð
ÞÀλ À 2
ð
ÞÁÁÁÀλ À m þ 1=m! and
Àλ
0
1: ð3:182Þ
Notice that (3.181) with (3.182) is an extension of binomial theorem (generalized
binomial theorem). Putting Àλ ¼ n, we have
n
m
¼
n!
n À m
ð
Þ!m!
and
n
0
¼ 1:
We rewrite (3.181) using gamma functions Γ(z) such that
1 þ x
ð
Þ
Àλ ¼
X 1
m¼0
Γ Àλ þ 1
ð
Þ
m!Γ Àλ À m þ 1
ð
Þ
x
m ,
ð3:183Þ
where Γ(z) is defined by integral representation as
Γ z
ð Þ ¼
Z 1
0
e
Àt t
zÀ1 dt Re z > 0
ð
Þ :
ð3:184Þ
Changing variables such that t ¼ u
2 , we have
3.6 Orbital Angular Momentum: Analytic Approach
95