New q--tangent and q--secant numbers

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.

This paper contains an open problem! A continued fraction!

Added August 25, 2000: Wow! Markus Fulmek proved the conjectured continued fraction expansion!

http://xxx.lanl.gov/abs/math.CO/0008180

This paper is available in the Tex, Dvi, PostScript format.

helmut@gauss.cam.wits.ac.za,

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