The 𝑞-tangent and 𝑞-secant numbers via basic Eulerian polynomials

Foata, Dominique; Han, Guo-Niu

doi:10.1090/S0002-9939-09-10144-2

The $q$-tangent and $q$-secant numbers via basic Eulerian polynomials
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by Dominique Foata and Guo-Niu Han PDF

Proc. Amer. Math. Soc. 138 (2010), 385-393 Request permission

Abstract:

The classical identity that relates Eulerian polynomials to tangent numbers together with the parallel result dealing with secant numbers is given a $q$-extension, both analytically and combinatorially. The analytic proof is based on a recent result by Shareshian and Wachs and the combinatorial one on the geometry of alternating permutations.

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