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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 SchlaefliS

Source Notebook

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:

In[1]:=
ResourceFunction["SchlaefliS"][3, 2.41]
Out[1]=

Evaluate Schläfli polynomials for various orders:

In[2]:=
Table[ResourceFunction["SchlaefliS"][n, z], {n, -4, 4}] // Expand
Out[2]=

Plot with respect to z:

In[3]:=
Plot[ResourceFunction["SchlaefliS"][Range[4], z] // Evaluate, {z, 0, 4}]
Out[3]=

Scope (3) 

Evaluate for complex arguments:

In[4]:=
ResourceFunction["SchlaefliS"][3, 2.1 I]
Out[4]=

Evaluate to high precision:

In[5]:=
N[ResourceFunction["SchlaefliS"][4, 9/2], 50]
Out[5]=

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

In[6]:=
ResourceFunction["SchlaefliS"][4, 4.500000000000000000000000]
Out[6]=

SchlaefliS threads elementwise over lists:

In[7]:=
ResourceFunction["SchlaefliS"][-2, {0.5, 1.0, 1.5}]
Out[7]=

Properties and Relations (5) 

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

In[8]:=
Table[n/2 ResourceFunction["SchlaefliS"][n, t] == t ResourceFunction["NeumannO"][n, t] - Cos[(n \[Pi])/2]^2 // Simplify, {n, -3, 3}]
Out[8]=

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

In[9]:=
Table[ResourceFunction["SchlaefliS"][n, z] == (n + 1 + (n - 1) (-1)^n) ResourceFunction[ "LommelS"][-Mod[n + 1, 2], n, z], {n, 0, 9}] // Simplify
Out[9]=

Verify a differential equation for the Schläfli polynomial:

In[10]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/fa0f62dd-9461-4052-9036-a3495d6ffd4f"]
Out[10]=

Verify a recurrence identity for the Schläfli polynomial:

In[11]:=
Table[ResourceFunction["SchlaefliS"][n + 1, z] + ResourceFunction["SchlaefliS"][n - 1, z] - (2 n)/ z ResourceFunction["SchlaefliS"][n, z] == 4/z Cos[(n \[Pi])/2]^2, {n, 0, 9}] // Simplify
Out[11]=

Verify Graf's formula for the Schläfli polynomial:

In[12]:=
Table[ResourceFunction["SchlaefliS"][n, z] == \[Pi] \!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = \(-n\)\), \(n\)]\((BesselJ[n, z] BesselY[k, z] - BesselJ[k, z] BesselY[n, z])\)\) // FullSimplify, {n, 0, 9}]
Out[12]=

Version History

  • 1.0.0 – 13 September 2021

Source Metadata

  • Citation:
    • Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 284–290, 1966.

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