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

\( O_0^{(\alpha)}(t)=\frac 1 t, \)
\( O_1^{(\alpha)}(t)=2\frac {\alpha+1}{t^2}, \)
\( O_2^{(\alpha)}(t)=\frac {2+\alpha}{t}+ 4\frac {(2+\alpha)(1+\alpha)}{t^3}, \)
\( O_3^{(\alpha)}(t)=2\frac {(1+\alpha)(3+\alpha)}{t^2}+ 8\frac {(1+\alpha)(2+\alpha)(3+\alpha)}{t^4}, \)
\( O_4^{(\alpha)}(t)=\frac {(1+\alpha)(4+\alpha)}{2t}+ 4\frac {(1+\alpha)(2+\alpha)(4+\alpha)}{t^3}+ 16\frac {(1+\alpha)(2+\alpha)(3+\alpha)(4+\alpha)}{t^5}. \)

A general form for the polynomial is

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

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 form

\( f(z)=\sum_{n=0} a_n J_{\alpha+n}(z)\, \)

for |z|<c compute

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

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

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

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 z 2\right)^s}. \)

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

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

the confluent hypergeometric function

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

and in particular

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

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. [3][not in citation given]) 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
Lommel polynomial
Hankel transform
Fourier–Bessel series‎

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

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