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. 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 .