In mathematics, the Neumann polynomials, introduced by Carl Neumann 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

, compute

and c is the distance of the nearest singularity of f(z).

Examples

An example is the extension

or the more general Sonine formula

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

Notes