# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

Compute the Schett polynomial

Contributed by:
Shenghui Yang

ResourceFunction["SchettPolynomial"][ computes the | |

ResourceFunction["SchettPolynomial"][ computes the | |

ResourceFunction["SchettPolynomial"][ expresses the coefficients of | |

ResourceFunction["SchettPolynomial"]["Reset"] clears the memory used for efficiency. |

Alois Schett introduced the recurrence polynomial starting from *S*_{0}=*x* and the recurrence relation *S*_{n}=D(*S*_{n-1}), where D(*x*)=*y*·*z*,D(*y*)=*z*·*x*,D(*z*)=*x*·*y* and the Leibniz rule applies.

The coefficients of the Schett polynomial are contained in the Taylor series of the Jacobi elliptic functions JacobiCN, JacobiSN and JacobiDN.

The coefficients of Schett polynomial are proven to count certain type of permutations.

The recurrence uses caching to speed up computation and avoid repetitive operation. Use SchettRecurrence["Reset"] to clean the memorization and release memory.

The fifth Schett polynomial:

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The fifth Schett polynomial with specified variables:

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Some coefficients of n^{th}Schett recurrence grows as rapidly as n!. For large index, it is convenient to use SparseArray to represent the polynomial:

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The Schett recurrence formula significantly speeds up the computation for high order terms in Taylor series for Jacobi elliptic functions:

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The sum of the coefficients in *S*_{n} is equal to *n*!:

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The coefficient of the Taylor series for JacobiSN is contained in the first column of the compact coefficient matrix for the Schett polynomial:

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The denominators in the Taylor series are factorials of odd numbers:

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The coefficient of Taylor series for JacobiCN is contained in the first row of the compact coefficient matrix:

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The denominators in the Taylor series are factorials of even numbers:

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The coefficient of Taylor series for JacobiDN is contained in the first row of the compact coefficient matrix:

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The denominators in the Taylor series are just factorials of even numbers:

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The coefficient matrix is not defined for indices less than 2:

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The (i,j)^{th} term in the coefficient matrix does not represent the coefficient of x^{i}y^{j}z^{n-i-j} in the corresponding polynomial:

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Instead, a general index conversion formula is applied: (i,j) → x^{2j-1}y^{2i-2}z^{n+4-2i-2j} for even n and (i,j) → x^{2j-2}y^{2i-1}z^{n+4-2i-2j} for odd n. For instance, 408 in S_{8} above occurs at {1,2} and {4,2}. The exponent for the term in the polynomial is:

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In this example one finds that 408 is the coefficient for x^{3}y^{0}z^{6} and x^{3}y^{6}z^{0}.

SchettPolynomial use memoization to speed up computation and the memory is not automatically released. Use "Reset" to free up memory space:

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Wolfram Language 12.3 (May 2021) or above

- 1.0.0 – 12 January 2024

This work is licensed under a Creative Commons Attribution 4.0 International License