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 f(z) from .
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.