Neumann polynomial

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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.[1]

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 .

Examples

Summarize
Perspective

An example is the extension

or the more general Sonine formula[2]

where is Gegenbauer's polynomial. Then,[citation needed][original research?]

the confluent hypergeometric function

and in particular

the index shift formula

the Taylor expansion (addition formula)

(cf.[3][failed verification]) and the expansion of the integral of the Bessel function,

are of the same type.

See also

Notes

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