Function Repository Resource:

# NeumannO

Evaluate the Neumann polynomial

Contributed by: Yury Brychkov and Jan Mangaldan
 ResourceFunction["NeumannO"][n,z] gives the Neumann polynomial On(z) .

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The Neumann polynomial is defined as: Neumann polynomials satisfy the generalized generating function relation .
Neumann polynomials satisfy the integral representation .
Neumann polynomials satisfy the differential equation Neumann polynomials are rational functions and not strictly polynomials.
ResourceFunction["NeumannO"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (3)

Evaluate numerically:

 In:= Out= Evaluate Neumann polynomials for various orders:

 In:= Out= Plot with respect to z:

 In:= Out= ### Scope (3)

Evaluate for complex arguments:

 In:= Out= Evaluate to high precision:

 In:= Out= The precision of the output tracks the precision of the input:

 In:= Out= In:= Out= ### Applications (3)

Define a function: Use NeumannO to expand a function in a Bessel function series:

 In:= Out= Compare the function with its Bessel series approximation:

 In:= Out= ### Properties and Relations (8)

Derivatives of Neumann polynomials are related to the polynomials themselves via :

 In:= Out= Neumann polynomials satisfy the differential equation :

 In:= Out= Neumann polynomials satisfy the recurrence identity :

 In:= Out= The Neumann polynomials have the limiting behavior given by :

 In:= Out= Neumann polynomials can be represented as the finite sum :

 In:= Out= The Neumann polynomials can be expressed in terms of HypergeometricPFQ through the formula :

 In:= Out= Neumann polynomials can be expressed in terms of the Lommel function:

 In:= Out= Neumann polynomials can be expressed in terms of the Schläfli polynomial:

 In:= Out= ## Version History

• 1.0.1 – 31 August 2021
• 1.0.0 – 06 December 2019