Neumann polynomial

In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case α = 0 {\displaystyle \alpha =0} , are a sequence of polynomials in 1 / t {\displaystyle 1/t} used to expand functions in term of Bessel functions.^[1]

The first few polynomials are

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

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

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

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

O 4 ( α ) ( t ) = ( 1 + α ) ( 4 + α ) 2 t + 4 ( 1 + α ) ( 2 + α ) ( 4 + α ) t 3 + 16 ( 1 + α ) ( 2 + α ) ( 3 + α ) ( 4 + α ) t 5 . {\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}}}.}

A general form for the polynomial is

O n ( α ) ( t ) = α + n 2 α ∑ k = 0 ⌊ n / 2 ⌋ ( − 1 ) n − k ( n − k ) ! k ! ( − α n − k ) ( 2 t ) n + 1 − 2 k , {\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},}

and they have the "generating function"

( z 2 ) α Γ ( α + 1 ) 1 t − z = ∑ n = 0 O n ( α ) ( t ) J α + n ( z ) , {\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),}

where J are Bessel functions.

To expand a function f in the form

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

for | t | < c {\displaystyle |t|<c} , compute

a n = Γ ( α + 1 ) 2 π i ∮ | t | = c ′ f ( t ) O n ( α ) ( t ) d t , {\displaystyle a_{n}={\frac {\Gamma (\alpha +1)}{2\pi i}}\oint _{|t|=c'}f(t)O_{n}^{(\alpha )}(t)\,dt,}

where c ′ < c {\displaystyle c'<c} and c is the distance of the nearest singularity of f(z) from z = 0 {\displaystyle z=0} .

An example is the extension

( 1 2 z ) s = Γ ( s ) ⋅ ∑ k = 0 ( − 1 ) k J s + 2 k ( z ) ( s + 2 k ) ( − s k ) , {\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},}

or the more general Sonine formula^[2]

e i γ z = Γ ( s ) ⋅ ∑ k = 0 i k C k ( s ) ( γ ) ( s + k ) J s + k ( z ) ( z 2 ) s . {\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}}}.}

where C k ( s ) {\displaystyle C_{k}^{(s)}} is Gegenbauer's polynomial. Then,^{[citation needed][original research?]}

( z 2 ) 2 k ( 2 k − 1 ) ! J s ( z ) = ∑ i = k ( − 1 ) i − k ( i + k − 1 2 k − 1 ) ( i + k + s − 1 2 k − 1 ) ( s + 2 i ) J s + 2 i ( z ) , {\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),}

∑ n = 0 t n J s + n ( z ) = e t z 2 t s ∑ j = 0 ( − z 2 t ) j j ! γ ( j + s , t z 2 ) Γ ( j + s ) = ∫ 0 ∞ e − z x 2 2 t z x t J s ( z 1 − x 2 ) 1 − x 2 s d x , {\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,}

the confluent hypergeometric function

M ( a , s , z ) = Γ ( s ) ∑ k = 0 ∞ ( − 1 t ) k L k ( − a − k ) ( t ) J s + k − 1 ( 2 t z ) ( t z ) s − k − 1 , {\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}}},}

and in particular

J s ( 2 z ) z s = 4 s Γ ( s + 1 2 ) π e 2 i z ∑ k = 0 L k ( − s − 1 / 2 − k ) ( i t 4 ) ( 4 i z ) k J 2 s + k ( 2 t z ) t z 2 s + k , {\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}}},}

the index shift formula

Γ ( ν − μ ) J ν ( z ) = Γ ( μ + 1 ) ∑ n = 0 Γ ( ν − μ + n ) n ! Γ ( ν + n + 1 ) ( z 2 ) ν − μ + n J μ + n ( z ) , {\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),}

the Taylor expansion (addition formula)

J s ( z 2 − 2 u z ) ( z 2 − 2 u z ) ± s = ∑ k = 0 ( ± u ) k k ! J s ± k ( z ) z ± s , {\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][failed verification]}) and the expansion of the integral of the Bessel function,

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

are of the same type.

• Bessel function
• Bessel polynomial
• Lommel polynomial
• Hankel transform
• Fourier–Bessel series