Function Repository Resource:

# SchlaefliS

Evaluate the Schläfli polynomial

Contributed by: Jan Mangaldan
 ResourceFunction["SchlaefliS"][n,z] gives the Schläfli polynomial Sn(z) .

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
ResourceFunction["SchlaefliS"] is defined as:
Schläfli polynomials are strictly rational functions and not polynomials.
ResourceFunction["SchlaefliS"] can be evaluated to arbitrary numerical precision.
ResourceFunction["SchlaefliS"] automatically threads over lists.

## Examples

### Basic Examples (3)

Evaluate numerically:

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Evaluate Schläfli polynomials for various orders:

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Plot with respect to z:

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### Scope (3)

Evaluate for complex arguments:

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Evaluate to high precision:

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The precision of the output tracks the precision of the input:

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SchlaefliS threads elementwise over lists:

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### Properties and Relations (5)

The Schläfli polynomial can be expressed in terms of the Neumann polynomial NeumannO:

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The Schläfli polynomial can be expressed in terms of the Lommel function LommelS:

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Verify a differential equation for the Schläfli polynomial:

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Verify a recurrence identity for the Schläfli polynomial:

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Verify Graf's formula for the Schläfli polynomial:

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## Version History

• 1.0.0 – 13 September 2021