Dyck graph

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.

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

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

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

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

where $c^{\prime }<c$ and c is the distance of the nearest singularity of $z^{-\alpha }f(z)$ from $z=0$.

Examples

An example is the extension

${\left(\frac{1}{2}z\right)}^s=\Gamma (s)\cdot \sum _{k=0}(-1{)}^k{J}_{s+2k}(z)(s+2k)(\genfrac{}{}{0ex}{}{-s}{k})$

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,Template:FactTemplate:Or

$\frac{{\left(\frac{z}{2}\right)}^{2k}}{(2k-1)!}{J}_s(z)=\sum _{i=k}(-1{)}^{i-k}(\genfrac{}{}{0ex}{}{i+k-1}{2k-1})(\genfrac{}{}{0ex}{}{i+k+s-1}{2k-1})(s+2i)J_{s+2i}(z),$

$\sum _{n=0}{t}^n{J}_{s+n}(z)=\frac{e^{\frac{tz}{2}}}{t^s}\sum _{j=0}\frac{{(-\frac{z}{2t})}^j}{j!}\frac{\gamma (j+s,\frac{tz}{2})}{\Gamma (j+s)}={\int }_0^{\mathrm{\infty }}{e}^{-\frac{z{x}^2}{2t}}\frac{zx}{t}\frac{J_s(z\sqrt{1-x^2})}{{\sqrt{1-x^2}}^s}dx,$

the confluent hypergeometric function

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

and in particular

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

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.Template:Verification failed) 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