In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case
, are a sequence of polynomials in
used to expand functions in term of Bessel functions.[1]
The first few polynomials are
![{\displaystyle O_{0}^{(\alpha )}(t)={\frac {1}{t)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/516fd507781641157f28aff57564bcad10f5df32)
![{\displaystyle O_{1}^{(\alpha )}(t)=2{\frac {\alpha +1}{t^{2))},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebae0012770cb495126196398223ab55d7407983)
![{\displaystyle O_{2}^{(\alpha )}(t)={\frac {2+\alpha }{t))+4{\frac {(2+\alpha )(1+\alpha )}{t^{3))},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31153c893ee1f6a8d49071ed8037bc8969787da7)
![{\displaystyle O_{3}^{(\alpha )}(t)=2{\frac {(1+\alpha )(3+\alpha )}{t^{2))}+8{\frac {(1+\alpha )(2+\alpha )(3+\alpha )}{t^{4))},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd00182b9aebdcb97681f84a9af06bac349cdfc2)
![{\displaystyle 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))}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/306ca93b8078719bcc46225cc5c5e568b1e66a9e)
A general form for the polynomial is
![{\displaystyle 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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5811873893a3d684471cd0496b2a07b23e32f4)
and they have the "generating function"
![{\displaystyle {\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b945a5cd14fc1cf51f697561c0885fd11b61103f)
where J are Bessel functions.
To expand a function f in the form
![{\displaystyle f(z)=\sum _{n=0}a_{n}J_{\alpha +n}(z)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d621ef164172bdfaf08c835fe40e9190ed338bfc)
for
, compute
![{\displaystyle 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)\,dz,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f25153c014311f6324142effc527ef896189e0)
where
and c is the distance of the nearest singularity of
from
.
Examples
An example is the extension
![{\displaystyle \left({\tfrac {1}{2))z\right)^{s}=\Gamma (s)\cdot \sum _{k=0}(-1)^{k}J_{s+2k}(z)(s+2k){-s \choose k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/317af413d973610c37ebbfea43d02bf2dd19b9d0)
or the more general Sonine formula[2]
![{\displaystyle 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))}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83a7210cfc7bf7366044d60475487d8eefe1668c)
where
is Gegenbauer's polynomial. Then,[citation needed][original research?]
![{\displaystyle {\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/474fedf86a424e981655d6e4021a850e3dc88eae)
![{\displaystyle \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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e339ee666e5875f6e0dc2e414f259299753b7e3)
the confluent hypergeometric function
![{\displaystyle 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))},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1017b7d17601d09e65272e29af52102c6eedd4ae)
and in particular
![{\displaystyle {\frac {J_{s}(2z)}{z^{s))}={\frac {4^{s}\Gamma \left(s+{\frac {1}{2))\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))},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/085b30599433bb092d632458e3a08f801bf67f59)
the index shift formula
![{\displaystyle \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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47de1a317e7dc58c9260d42724bfa3a95cabea60)
the Taylor expansion (addition formula)
![{\displaystyle {\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))},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3498285a600bd5dd5bae9ae80d23ce8a5824607e)
(cf.[3][failed verification]) and the expansion of the integral of the Bessel function,
![{\displaystyle \int J_{s}(z)dz=2\sum _{k=0}J_{s+2k+1}(z),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd37326fb1c4b1783436127758e8cab2413523b)
are of the same type.