Multivariable tangent and secant \(q\)-derivative polynomials

zbMATH Open1270.05006arXiv1304.2486MaRDI QIDQ1946758

Guo-Niu Han, Dominique Foata

Publication date: 15 April 2013

Published in: Moscow Journal of Combinatorics and Number Theory (Search for Journal in Brave)

Abstract: The derivative polynomials introduced by Knuth and Buckholtz in their calculations of the tangent and secant numbers are extended to a multivariable $q$--environment. The $n$-th $q$-derivatives of the classical $q$-tangent and $q$-secant are each given two polynomial expressions. The first polynomial expression is indexed by triples of integers, the second by compositions of integers. The functional relation between those two classes is fully given by means of combinatorial techniques. Moreover, those polynomials are proved to be generating functions for so-called $t$-permutations by multivariable statistics. By giving special values to those polynomials we recover classical $q$-polynomials such as the Carlitz $q$-Eulerian polynomials and the $(t,q)$-tangent and -secant analogs recently introduced. They also provide $q$-analogs for the Springer numbers. Finally, the $t$-compositions used in this paper furnish a combinatorial interpretation to one of the Fibonacci triangles.

Full work available at URL: https://arxiv.org/abs/1304.2486

zbMATH Keywords

inversion numberalternating permutations\(q\)-tangent numbersinverse major index\(q\)-derivative polynomials\(q\)-second numbers\(q\)-Springer numbers\(s\)-compositions\(t\)-compositions\(t\)-permutations\(tq\)-secant numbers\(tq\)-tangent numbers

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) (q)-calculus and related topics (05A30) Bernoulli and Euler numbers and polynomials (11B68) Exponential and trigonometric functions (33B10)

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