International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 12, Pages 825-838
doi:10.1155/S0161171200004439
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
Abstract
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.