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
:$O_0^(t)=\frac 1 t,$
:$O_1^(t)=2\frac ,$
:$O_2^(t)=\frac + 4\frac ,$
:$O_3^(t)=2\frac + 8\frac ,$
:$O_4^(t)=\frac + 4\frac + 16\frac .$

A general form for the polynomial is
:$O_n^(t)= \frac \sum_^ (-1)^\frac \left(\frac 2 t \right)^,$
and they have the "generating function"
:$\frac \frac 1 = \sum_O_n^(t) J_(z),$
where ''J'' are Bessel functions.

To expand a function ''f'' in the form
:$f(z)=\sum_ a_n J_(z)\,$
for , z, , compute
:$a_n=\frac 1 \oint_ \fracf(z) O_n^(z)\,dz,$
where c' and ''c'' is the distance of the nearest singularity of $z^ f(z)$ from $z=0$.

Examples

An example is the extension
:$\left(\tfracz\right)^s= \Gamma(s)\cdot\sum_(-1)^k J_(z)(s+2k),$
or the more general Sonine formula II.7.10.1, p.64
:$e^= \Gamma(s)\cdot\sum_i^k C_k^(\gamma)(s+k)\frac.$
where $C_k^$ is Gegenbauer's polynomial. Then,
:$\fracJ_s(z)= \sum_(-1)^(s+2i)J_(z),$
:$\sum_ t^n J_(z)= \frac \sum_\frac\frac= \int_0^\infty e^\frac \frac\,dx,$
the confluent hypergeometric function
:$M(a,s,z)= \Gamma (s) \sum_^\infty \left(-\frac\right)^k L_k^(t) \frac,$
and in particular
:$\frac= \frace^\sum_L_k^\left(\frac4\right)(4 i z)^k \frac,$
the index shift formula
:$\Gamma(\nu-\mu) J_\nu(z)= \Gamma(\mu+1) \sum_\frac \left(\frac z 2\right)^J_(z),$
the Taylor expansion (addition formula)
:$\frac= \sum_\frac\frac,$
(cf.) and the expansion of the integral of the Bessel function,
:$\int J_s(z)dz= 2 \sum_ J_{s+2k+1}(z),$
are of the same type.

See also

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

Notes

Polynomials
Special functions