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. 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"
… excerpt ends here. Continue reading the full article.
