International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 12, Pages 825-838

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

Helmut Prodinger

The John Knopfmacher Centre for Applicable Analysis and Number Theory, Department of Mathematics, University of the Witwatersrand, P.O. Wits, Johannesburg 2050, South Africa

Received 1 November 1999

Copyright © 2000 Helmut Prodinger. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Up-down permutations are counted by tangent (respectively, 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 à la Andrews, Foata, Gessel are also discussed.