In mathematics, a Neumann polynomial, introduced by Carl Neumann for the special case
α
=
0
, is a polynomial in 1/z used to expand functions in term of Bessel functions.
The first few polynomials are
O
0
(
α
)
(
t
)
=
1
t
,
O
1
(
α
)
(
t
)
=
2
α
+
1
t
2
,
O
2
(
α
)
(
t
)
=
2
+
α
t
+
4
(
2
+
α
)
(
1
+
α
)
t
3
,
O
3
(
α
)
(
t
)
=
2
(
1
+
α
)
(
3
+
α
)
t
2
+
8
(
1
+
α
)
(
2
+
α
)
(
3
+
α
)
t
4
,
O
4
(
α
)
(
t
)
=
(
1
+
α
)
(
4
+
α
)
2
t
+
4
(
1
+
α
)
(
2
+
α
)
(
4
+
α
)
t
3
+
16
(
1
+
α
)
(
2
+
α
)
(
3
+
α
)
(
4
+
α
)
t
5
.
A general form for the polynomial is
O
n
(
α
)
(
t
)
=
α
+
n
2
α
∑
k
=
0
⌊
n
/
2
⌋
(
−
1
)
n
−
k
(
n
−
k
)
!
k
!
(
−
α
n
−
k
)
(
2
t
)
n
+
1
−
2
k
,
they have the generating function
(
z
2
)
α
Γ
(
α
+
1
)
1
t
−
z
=
∑
n
=
0
O
n
(
α
)
(
t
)
J
α
+
n
(
z
)
,
where J are Bessel functions.
To expand a function f in form
f
(
z
)
=
∑
n
=
0
a
n
J
α
+
n
(
z
)
for
|
z
|
<
c
compute
a
n
=
1
2
π
i
∮
|
z
|
=
c
′
Γ
(
α
+
1
)
(
z
2
)
α
f
(
z
)
O
n
(
α
)
(
z
)
d
z
,
where
c
′
<
c
and c is the distance of the nearest singularity of
z
−
α
f
(
z
)
from
z
=
0
.
An example is the extension
(
1
2
z
)
s
=
Γ
(
s
)
⋅
∑
k
=
0
(
−
1
)
k
J
s
+
2
k
(
z
)
(
s
+
2
k
)
(
−
s
k
)
or the more general Sonine formula
e
i
γ
z
=
Γ
(
s
)
⋅
∑
k
=
0
i
k
C
k
(
s
)
(
γ
)
(
s
+
k
)
J
s
+
k
(
z
)
(
z
2
)
s
.
where
C
k
(
s
)
is Gegenbauer's polynomial. Then,
(
z
2
)
2
k
(
2
k
−
1
)
!
J
s
(
z
)
=
∑
i
=
k
(
−
1
)
i
−
k
(
i
+
k
−
1
2
k
−
1
)
(
i
+
k
+
s
−
1
2
k
−
1
)
(
s
+
2
i
)
J
s
+
2
i
(
z
)
,
∑
n
=
0
t
n
J
s
+
n
(
z
)
=
e
t
z
2
t
s
∑
j
=
0
(
−
z
2
t
)
j
j
!
γ
(
j
+
s
,
t
z
2
)
Γ
(
j
+
s
)
=
∫
0
∞
e
−
z
x
2
2
t
z
x
t
J
s
(
z
1
−
x
2
)
1
−
x
2
s
d
x
,
the confluent hypergeometric function
M
(
a
,
s
,
z
)
=
Γ
(
s
)
∑
k
=
0
∞
(
−
1
t
)
k
L
k
(
−
a
−
k
)
(
t
)
J
s
+
k
−
1
(
2
t
z
)
(
t
z
)
s
−
k
−
1
and in particular
J
s
(
2
z
)
z
s
=
4
s
Γ
(
s
+
1
2
)
π
e
2
i
z
∑
k
=
0
L
k
(
−
s
−
1
/
2
−
k
)
(
i
t
4
)
(
4
i
z
)
k
J
2
s
+
k
(
2
t
z
)
t
z
2
s
+
k
,
the index shift formula
Γ
(
ν
−
μ
)
J
ν
(
z
)
=
Γ
(
μ
+
1
)
∑
n
=
0
Γ
(
ν
−
μ
+
n
)
n
!
Γ
(
ν
+
n
+
1
)
(
z
2
)
ν
−
μ
+
n
J
μ
+
n
(
z
)
,
the Taylor expansion (addition formula)
J
s
(
z
2
−
2
u
z
)
(
z
2
−
2
u
z
)
±
s
=
∑
k
=
0
(
±
u
)
k
k
!
J
s
±
k
(
z
)
z
±
s
(cf.) and the expansion of the integral of the Bessel function
∫
J
s
(
z
)
d
z
=
2
∑
k
=
0
J
s
+
2
k
+
1
(
z
)
are of the same type.
