In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case
α
=
0
{\displaystyle \alpha =0}
, are a sequence of polynomials in
1
/
t
{\displaystyle 1/t}
used to expand functions in term of Bessel functions.
The first few polynomials are
O
0
(
α
)
(
t
)
=
1
t
,
{\displaystyle O_{0}^{(\alpha )}(t)={\frac {1}{t}},}
O
1
(
α
)
(
t
)
=
2
α
+
1
t
2
,
{\displaystyle O_{1}^{(\alpha )}(t)=2{\frac {\alpha +1}{t^{2}}},}
O
2
(
α
)
(
t
)
=
2
+
α
t
+
4
(
2
+
α
)
(
1
+
α
)
t
3
,
{\displaystyle O_{2}^{(\alpha )}(t)={\frac {2+\alpha }{t}}+4{\frac {(2+\alpha )(1+\alpha )}{t^{3}}},}
O
3
(
α
)
(
t
)
=
2
(
1
+
α
)
(
3
+
α
)
t
2
+
8
(
1
+
α
)
(
2
+
α
)
(
3
+
α
)
t
4
,
{\displaystyle O_{3}^{(\alpha )}(t)=2{\frac {(1+\alpha )(3+\alpha )}{t^{2}}}+8{\frac {(1+\alpha )(2+\alpha )(3+\alpha )}{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
.
{\displaystyle O_{4}^{(\alpha )}(t)={\frac {(1+\alpha )(4+\alpha )}{2t}}+4{\frac {(1+\alpha )(2+\alpha )(4+\alpha )}{t^{3}}}+16{\frac {(1+\alpha )(2+\alpha )(3+\alpha )(4+\alpha )}{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
,
{\displaystyle O_{n}^{(\alpha )}(t)={\frac {\alpha +n}{2\alpha }}\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{n-k}{\frac {(n-k)!}{k!}}{-\alpha \choose n-k}\left({\frac {2}{t}}\right)^{n+1-2k},}
and they have the "generating function"
(
z
2
)
α
Γ
(
α
+
1
)
1
t
−
z
=
∑
n
=
0
O
n
(
α
)
(
t
)
J
α
+
n
(
z
)
,
{\displaystyle {\frac {\left({\frac {z}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}{\frac {1}{t-z}}=\sum _{n=0}O_{n}^{(\alpha )}(t)J_{\alpha +n}(z),}
where J are Bessel functions.
To expand a function f in the form
f
(
z
)
=
(
2
z
)
α
∑
n
=
0
a
n
J
α
+
n
(
z
)
{\displaystyle f(z)=\left({\frac {2}{z}}\right)^{\alpha }\sum _{n=0}a_{n}J_{\alpha +n}(z)\,}
for
|
t
|
<
c
{\displaystyle |t|<c}
, compute
a
n
=
Γ
(
α
+
1
)
2
π
i
∮
|
t
|
=
c
′
f
(
t
)
O
n
(
α
)
(
t
)
d
t
,
{\displaystyle a_{n}={\frac {\Gamma (\alpha +1)}{2\pi i}}\oint _{|t|=c'}f(t)O_{n}^{(\alpha )}(t)\,dt,}
where
c
′
<
c
{\displaystyle c'<c}
and c is the distance of the nearest singularity of f(z) from
z
=
0
{\displaystyle z=0}
.
Examples
An example is the extension
(
1
2
z
)
s
=
Γ
(
s
)
⋅
∑
k
=
0
(
−
1
)
k
J
s
+
2
k
(
z
)
(
s
+
2
k
)
(
−