Neumann polynomial
Introduction 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.The first few polynomials are:::::A general form for the polynomial is:and they have the "generating function" :where ''J'' are Bessel functions.To expand a function ''f'' in the form :for , compute:where  and ''c'' is the distance of the nearest singularity of  from .
Examples An example is the extension:or the more general Sonine formula:where  is Gegenbauer's polynomial. Then,::the confluent hypergeometric function:and in particular:the index shift formula:the Taylor expansion (addition formula):(cf.) and the expansion of the integral of the Bessel function,:are of the same type.
See also *Bessel function*Bessel polynomial*Lommel polynomial*Hankel transform*Fourier–Bessel series*Schläfli-polynomial
Notes