Enumeration formulas for generalized q-Euler numbers

Abstract

We find an enumeration formula for a $(t,q)$-Euler number which is a generalization of the $q$-Euler number introduced by Han, Randrianarivony, and Zeng. We also give a combinatorial expression for the $(t,q)$-Euler number and find another formula when $t=\pm q^r$ for any integer $r$. Special cases of our latter formula include the formula of the $q$-Euler number recently found by Josuat-Vergès and Touchard-Riordan's formula.

21 pages, 12 figures

References in corpus (3)

• The $q$-tangent and $q$-secant numbers via continued fractions
• A continued fraction expansion for a q-tangent function: An elementary proof
• On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs