Neumann polynomial

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

The first few polynomials are

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

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

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

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

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

A general form for the polynomial is

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

and they have the "generating function"

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

where J are Bessel functions.

To expand a function f in the form

$f(z)=\left({\frac {2}{z}}\right)^{\alpha }\sum _{n=0}a_{n}J_{\alpha n}(z)\,$

for $|t|<c$, compute

$a_{n}={\frac {\Gamma (\alpha 1)}{2\pi i}}\oint _{|t|=c'}f(t)O_{n}^{(\alpha )}(t)\,dt,$

where $c'<c$ and c is the distance of the nearest singularity of f(z) from $z=0$.

Examples

An example is the extension

$\left({\tfrac {1}{2}}z\right)^{s}=\Gamma (s)\cdot \sum _{k=0}(-1)^{k}J_{s 2k}(z)(s 2k){-s \choose k},$

or the more general Sonine formula

$e^{i\gamma z}=\Gamma (s)\cdot \sum _{k=0}i^{k}C_{k}^{(s)}(\gamma )(s k){\frac {J_{s k}(z)}{\left({\frac {z}{2}}\right)^{s}}}.$

where $C_{k}^{(s)}$ is Gegenbauer's polynomial. Then,

${\frac {\left({\frac {z}{2}}\right)^{2k}}{(2k-1)!}}J_{s}(z)=\sum _{i=k}(-1)^{i-k}{i k-1 \choose 2k-1}{i k s-1 \choose 2k-1}(s 2i)J_{s 2i}(z),$

$\sum _{n=0}t^{n}J_{s n}(z)={\frac {e^{\frac {tz}{2}}}{t^{s}}}\sum _{j=0}{\frac {\left(-{\frac {z}{2t}}\right)^{j}}{j!}}{\frac {\gamma \left(j s,{\frac {tz}{2}}\right)}{\,\Gamma (j s)}}=\int _{0}^{\infty }e^{-{\frac {zx^{2}}{2t}}}{\frac {zx}{t}}{\frac {J_{s}(z{\sqrt {1-x^{2}}})}{{\sqrt {1-x^{2}}}^{s}}}\,dx,$

the confluent hypergeometric function

$M(a,s,z)=\Gamma (s)\sum _{k=0}^{\infty }\left(-{\frac {1}{t}}\right)^{k}L_{k}^{(-a-k)}(t){\frac {J_{s k-1}\left(2{\sqrt {tz}}\right)}{({\sqrt {tz}})^{s-k-1}}},$

and in particular

${\frac {J_{s}(2z)}{z^{s}}}={\frac {4^{s}\Gamma \left(s {\frac {1}{2}}\right)}{\sqrt {\pi }}}e^{2iz}\sum _{k=0}L_{k}^{(-s-1/2-k)}\left({\frac {it}{4}}\right)(4iz)^{k}{\frac {J_{2s k}\left(2{\sqrt {tz}}\right)}{{\sqrt {tz}}^{2s k}}},$

the index shift formula

$\Gamma (\nu -\mu )J_{\nu }(z)=\Gamma (\mu 1)\sum _{n=0}{\frac {\Gamma (\nu -\mu n)}{n!\Gamma (\nu n 1)}}\left({\frac {z}{2}}\right)^{\nu -\mu n}J_{\mu n}(z),$

the Taylor expansion (addition formula)

${\frac {J_{s}\left({\sqrt {z^{2}-2uz}}\right)}{\left({\sqrt {z^{2}-2uz}}\right)^{\pm s}}}=\sum _{k=0}{\frac {(\pm u)^{k}}{k!}}{\frac {J_{s\pm k}(z)}{z^{\pm s}}},$

(cf.) and the expansion of the integral of the Bessel function,

$\int J_{s}(z)dz=2\sum _{k=0}J_{s 2k 1}(z),$

are of the same type.

See also

• Bessel function
• Bessel polynomial
• Lommel polynomial
• Hankel transform
• Fourier–Bessel series
• Schläfli-polynomial