Neumann polynomial

In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case formula_1, are a sequence of polynomials in formula_2 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 formula_11, compute
where formula_13 and "c" is the distance of the nearest singularity of formula_14 from formula_15.
Examples.
An example is the extension
or the more general Sonine formula
where formula_18 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.