Neumann polynomial

In mathematics, a Neumann polynomial, introduced by Carl Neumann for the special case $\alpha =0$, is a polynomial in 1/z used to expand functions in term of Bessel functions.^[1]

The first few polynomials are

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

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

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

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

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

A general form for the polynomial is

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

they have the generating function

${\frac {\left({\frac z2}\right)^{\alpha }}{\Gamma (\alpha +1)}}{\frac 1{t-z}}=\sum _{{n=0}}O_{n}^{{(\alpha )}}(t)J_{{\alpha +n}}(z),$

where J are Bessel functions.

To expand a function f in form

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

for $|z|<c$ compute

$a_{n}={\frac 1{2\pi i}}\oint _{{|z|=c'}}{\frac {\Gamma (\alpha +1)}{\left({\frac z2}\right)^{\alpha }}}f(z)O_{n}^{{(\alpha )}}(z){\mathrm d}z,$

where $c'<c$ and c is the distance of the nearest singularity of $z^{{-\alpha }}f(z)$ 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^[2]

$e^{{i\gamma z}}=\Gamma (s)\cdot \sum _{{k=0}}i^{k}C_{k}^{{(s)}}(\gamma )(s+k){\frac {J_{{s+k}}(z)}{\left({\frac z2}\right)^{s}}}.$

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

${\frac {\left({\frac z2}\right)^{{2k}}}{(2k-1)!}}J_{s}(z)=\sum _{{i=k}}(-1)^{{i-k}}{i+k-1 \choose 2k-1}{i+k+s-1 \choose 2k-1}(s+2i)J_{{s+2i}}(z),$

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

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

${\frac {J_{s}(2z)}{z^{s}}}={\frac {4^{s}\Gamma \left(s+{\frac 12}\right)}{{\sqrt \pi }}}e^{{2iz}}\sum _{{k=0}}L_{k}^{{(-s-1/2-k)}}\left({\frac {it}4}\right)(4iz)^{k}{\frac {J_{{2s+k}}\left(2{\sqrt {tz}}\right)}{{\sqrt {tz}}^{{2s+k}}}},$

the index shift formula

$\Gamma (\nu -\mu )J_{\nu }(z)=\Gamma (\mu +1)\sum _{{n=0}}{\frac {\Gamma (\nu -\mu +n)}{n!\Gamma (\nu +n+1)}}\left({\frac z2}\right)^{{\nu -\mu +n}}J_{{\mu +n}}(z),$

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.^[3]) 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

Notes

1. Abramowitz and Stegun, p. 363, 9.1.82 ff.
2. Erdélyi et al. 1955 II.7.10.1, p.64
3. Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "8.515.1.". In Zwillinger, Daniel; Moll, Victor Hugo. Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. p. 944. ISBN 0-12-384933-0. LCCN 2014010276. ISBN 978-0-12-384933-5.