Combinatorics of geometrically distributed random variables: New q-tangent and q-secant numbers

1999/10/19 by Helmut Prodinger, Prodinger, Helmut · 1 citation
Mathematics · #05A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15

paper · pdf · doi:10.48550/arxiv.math/9910096

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arxiv created 1999/10/19 · openalex publication_date 1999/10/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all coincide with the classical version. In this way, we get some new q-tangent and q-secant functions. Some of them also have nice continued fraction expansions; in one particular case, we could not find a proof for it. Divisibility results a la Andrews/Foata/Gessel are also discussed.

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