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

TL;DR

This paper introduces new q-tangent and q-secant functions derived from geometric distributions, exploring their properties, continued fractions, and divisibility results, thus extending classical combinatorial concepts to a probabilistic setting.

Contribution

It develops novel q-analogues of tangent and secant numbers based on geometric distributions, connecting combinatorics, probability, and special functions.

Findings

New q-tangent and q-secant functions with combinatorial interpretations.

Some functions have elegant continued fraction expansions.

Divisibility properties analogous to classical results are established.

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.