Neumann polynomial

In mathematics, a Neumanns 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

$O_0^{(\alpha)}(t)=\frac 1 t,$

$O_1^{(\alpha)}(t)=2\frac {\alpha%2B1}{t^2},$

$O_2^{(\alpha)}(t)=\frac {2%2B\alpha}{t}%2B 4\frac {(2%2B\alpha)(1%2B\alpha)}{t^3},$

$O_3^{(\alpha)}(t)=2\frac {(1%2B\alpha)(3%2B\alpha)}{t^2}%2B 8\frac {(1%2B\alpha)(2%2B\alpha)(3%2B\alpha)}{t^4},$

$O_4^{(\alpha)}(t)=\frac {(1%2B\alpha)(4%2B\alpha)}{2t}%2B 4\frac {(1%2B\alpha)(2%2B\alpha)(4%2B\alpha)}{t^3}%2B 16\frac {(1%2B\alpha)(2%2B\alpha)(3%2B\alpha)(4%2B\alpha)}{t^5}.$

A general form for the polynomial is

$O_n^{(\alpha)}(t)= \frac{\alpha%2Bn}{2\alpha} \sum_{k=0}^{\lfloor n/2\rfloor} (-1)^{n-k}\frac {(n-k)!} {k!} {-\alpha \choose n-k}\left(\frac 2 t \right)^{n%2B1-2k},$

they have the generating function

$\frac{\left(\frac z 2 \right)^\alpha} {\Gamma(\alpha%2B1)} \frac 1 {t-z}= \sum_{n=0}O_n^{(\alpha)}(t) J_{\alpha%2Bn}(z),$

where J are Bessel functions.

To expand a function f in form

$f(z)=\sum_{n=0} a_n J_{\alpha%2Bn}(z)\,$

for $|z|<c$ compute

$a_n=\frac 1 {2 \pi i} \oint_{|z|=c'} \frac{\Gamma(\alpha%2B1)}{\left(\frac z 2\right)^\alpha}f(z) O_n^{(\alpha)}(z)\mathrm d z,$

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

$\left(\tfrac{1}{2}z\right)^s= \Gamma(s)\cdot\sum_{k=0}(-1)^k J_{s%2B2k}(z)(s%2B2k){-s \choose k}$

or the more general Sonine formula^[2]

$e^{i \gamma z}= \Gamma(s)\cdot\sum_{k=0}i^k C_k^{(s)}(\gamma)(s%2Bk)\frac{J_{s%2Bk}(z)}{\left(\frac z 2\right)^s}.$

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

$\frac{\left(\frac z 2\right)^{2k}}{(2k-1)!}J_s(z)= \sum_{i=k}(-1)^{i-k}{i%2Bk-1\choose 2k-1}{i%2Bk%2Bs-1\choose 2k-1}(s%2B2i)J_{s%2B2i}(z),$

$\sum_{n=0} t^n J_{s%2Bn}(z)= \frac{e^{\frac{t z}2}}{t^s} \sum_{j=0}\frac{\left(-\frac{z}{2t}\right)^j}{j!}\frac{\gamma \left(j%2Bs,\frac{t z}{2}\right)}{\,\Gamma (j%2Bs)}= \int_0^\infty e^{-\frac{z x^2}{2 t}}\frac {z x}{t} \frac{J_s(z\sqrt{1-x^2})}{\sqrt{1-x^2}^s}\,dx,$

the confluent hypergeometric function

$M(a,s,z)= \Gamma (s) \sum_{k=0}^\infty \left(-\frac{1}{t}\right)^k L_k^{(-a-k)}(t) \frac{J_{s%2Bk-1}\left(2 \sqrt{t z}\right)}{(\sqrt{t z})^{s-k-1}}$

and in particular

$\frac{J_s(2 z)}{z^s}= \frac{4^s \Gamma\left(s%2B\frac12\right)}{\sqrt\pi}e^{2 i z}\sum_{k=0}L_k^{(-s-1/2-k)}\left(\frac{it}4\right)(4 i z)^k \frac{J_{2s%2Bk}\left(2\sqrt{t z}\right)}{\sqrt{t z}^{2s%2Bk}},$

the index shift formula

$\Gamma(\nu-\mu) J_\nu(z)= \Gamma(\mu%2B1) \sum_{n=0}\frac{\Gamma(\nu-\mu%2Bn)}{n!\Gamma(\nu%2Bn%2B1)} \left(\frac z 2\right)^{\nu-\mu%2Bn}J_{\mu%2Bn}(z),$

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

$\int J_s(z)dz= 2 \sum_{k=0} J_{s%2B2k%2B1}(z)$

are of the same type.

Notes

^ Abramowitz and Stegun, p. 363, 9.1.82 ff.
^ Erdélyi et al. 1955 II.7.10.1, p.64
^ I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжи); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Equation 8.515.1

Neumann polynomial

Examples

See also

Notes