Enumeration formulas for generalized q-Euler numbers

TL;DR

This paper introduces a new enumeration formula for a generalized $(t,q)$-Euler number, extending previous $q$-Euler numbers, and provides combinatorial expressions and special case formulas, connecting to known results.

Contribution

It presents a novel enumeration formula for the $(t,q)$-Euler number and derives combinatorial and special case formulas, expanding the understanding of $q$-Euler numbers.

Findings

Derived a general enumeration formula for $(t,q)$-Euler numbers.

Provided a combinatorial expression for the $(t,q)$-Euler number.

Connected special cases to known $q$-Euler and Riordan formulas.

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-Verg\`es and Touchard-Riordan's formula.