Neumann polynomial

In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case ${\displaystyle \alpha =0}$, are a sequence of polynomials in ${\displaystyle 1/t}$ used to expand functions in term of Bessel functions.^[1]

The first few polynomials are

${\displaystyle O_{0}^{(\alpha )}(t)={\frac {1}{t}},}$

${\displaystyle O_{1}^{(\alpha )}(t)=2{\frac {\alpha +1}{t^{2}}},}$

${\displaystyle O_{2}^{(\alpha )}(t)={\frac {2+\alpha }{t}}+4{\frac {(2+\alpha )(1+\alpha )}{t^{3}}},}$

${\displaystyle O_{3}^{(\alpha )}(t)=2{\frac {(1+\alpha )(3+\alpha )}{t^{2}}}+8{\frac {(1+\alpha )(2+\alpha )(3+\alpha )}{t^{4}}},}$

${\displaystyle O_{4}^{(\alpha )}(t)={\frac {(1+\alpha )(4+\alpha )}{2t}}+4{\frac {(1+\alpha )(2+\alpha )(4+\alpha )}{t^{3}}}+16{\frac {(1+\alpha )(2+\alpha )(3+\alpha )(4+\alpha )}{t^{5}}}.}$

A general form for the polynomial is

${\displaystyle O_{n}^{(\alpha )}(t)={\frac {\alpha +n}{2\alpha }}\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{n-k}{\frac {(n-k)!}{k!}}{-\alpha \choose n-k}\left({\frac {2}{t}}\right)^{n+1-2k},}$

and they have the "generating function"

${\displaystyle {\frac {\left({\frac {z}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}{\frac {1}{t-z}}=\sum _{n=0}O_{n}^{(\alpha )}(t)J_{\alpha +n}(z),}$

where J are Bessel functions.

To expand a function f in the form

${\displaystyle f(z)=\sum _{n=0}a_{n}J_{\alpha +n}(z)\,}$

for ${\displaystyle |z|<c}$, compute

${\displaystyle a_{n}={\frac {1}{2\pi i}}\oint _{|z|=c'}{\frac {\Gamma (\alpha +1)}{\left({\frac {z}{2}}\right)^{\alpha }}}f(z)O_{n}^{(\alpha )}(z)\,dz,}$

where ${\displaystyle c'<c}$ and c is the distance of the nearest singularity of ${\displaystyle z^{-\alpha }f(z)}$ from ${\displaystyle z=0}$.