polinomio de Neumann

En matemáticas, los polinomios de Neumann , introducidos por Carl Neumann para el caso especial${\displaystyle \alpha =0}$, son una secuencia de polinomios en${\displaystyle 1/t}$utilizado para expandir funciones en términos de funciones de Bessel . ^{[ 1 ]}

Los primeros polinomios son

${\displaystyle O_{0}^{(\alpha )}(t)={\frac {1}{t}},}$

${\displaystyle O_{1}^{(\alpha )}(t)=2{\frac {\alpha +1}{t^{2}}},}$

${\displaystyle O_{2}^{(\alpha )}(t)={\frac {2+\alpha }{t}}+4{\frac {(2+\alpha )(1+\alpha )}{t^{3}}},}$

${\displaystyle O_{3}^{(\alpha )}(t)=2{\frac {(1+\alpha )(3+\alpha )}{t^{2}}}+8{\frac {(1+\alpha )(2+\alpha )(3+\alpha )}{t^{4}}},}$

${\displaystyle O_{4}^{(\alpha )}(t)={\frac {(1+\alpha )(4+\alpha )}{2t}}+4{\frac {(1+\alpha )(2+\alpha )(4+\alpha )}{t^{3}}}+16{\frac {(1+\alpha )(2+\alpha )(3+\alpha )(4+\alpha )}{t^{5}}}.}$

Una forma general para el polinomio es

${\displaystyle O_{n}^{(\alpha )}(t)={\frac {\alpha +n}{2\alpha }}\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{nk}{\frac {(nk)!}{k!}}{-\alpha \choose nk}\left({\frac {2}{t}}\right)^{n+1-2k},}$

y tienen la "función generadora"

${\displaystyle {\frac {\left({\frac {z}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}{\frac {1}{tz}}=\sum _{n=0}O_{n}^{(\alpha )}(t)J_{\alpha +n}(z),}$

donde J son funciones de Bessel .

Para expandir una función f en la forma

${\displaystyle f(z)=\left({\frac {2}{z}}\right)^{\alpha }\sum _{n=0}a_{n}J_{\alpha +n}(z)\,}$

para${\displaystyle |t|<c}$, calcular

${\displaystyle a_{n}={\frac {\Gamma (\alpha +1)}{2\pi i}}\oint _{|t|=c'}f(t)O_{n}^{(\alpha )}(t)\,dt,}$

dónde${\displaystyle c'<c}$y c es la distancia de la singularidad más cercana de f(z) desde${\displaystyle z=0}$.

Un ejemplo es la extensión

${\displaystyle \left({\tfrac {1}{2}}z\right)^{s}=\Gamma (s)\cdot \sum _{k=0}(-1)^{k}J_{s+2k}(z)(s+2k){-s \choose k},}$

o la fórmula Sonine más general ^{[ 2 ]}

${\displaystyle e^{i\gamma z}=\Gamma (s)\cdot \sum _{k=0}i^{k}C_{k}^{(s)}(\gamma )(s+k){\frac {J_{s+k}(z)}{\left({\frac {z}{2}}\right)^{s}}}.}$

dónde${\displaystyle C_{k}^{(s)}}$es el polinomio de Gegenbauer . Entonces,

${\displaystyle {\frac {\left({\frac {z}{2}}\right)^{2k}}{(2k-1)!}}J_{s}(z)=\sum _{i=k}(-1)^{i-k}{i+k-1 \choose 2k-1}{i+k+s-1 \choose 2k-1}(s+2i)J_{s+2i}(z),}$

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

la función hipergeométrica confluente

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

y en particular

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

la fórmula de desplazamiento del índice

${\displaystyle \Gamma (\nu -\mu )J_{\nu }(z)=\Gamma (\mu +1)\sum _{n=0}{\frac {\Gamma (\nu -\mu +n)}{n!\Gamma (\nu +n+1)}}\left({\frac {z}{2}}\right)^{\nu -\mu +n}J_{\mu +n}(z),}$

La expansión de Taylor (fórmula de adición)

${\displaystyle {\frac {J_{s}\left({\sqrt {z^{2}-2uz}}\right)}{\left({\sqrt {z^{2}-2uz}}\right)^{\pm s}}}=\sum _{k=0}{\frac {(\pm u)^{k}}{k!}}{\frac {J_{s\pm k}(z)}{z^{\pm s}}},}$

(cf. ^{[ 3 ]} ) y la expansión de la integral de la función de Bessel,

${\displaystyle \int J_{s}(z)dz=2\sum _{k=0}J_{s+2k+1}(z),}$

son del mismo tipo.

• función de Bessel
• polinomio de Bessel
• polinomio de Lommel
• Transformación de Hankel
• series de Fourier-Bessel