Arithmetic properties of generalized Euler numbers

TL;DR

This paper investigates divisibility properties of a q-analog of generalized Euler numbers, extending classical results and exploring their combinatorial and algebraic structures.

Contribution

It introduces a q-analog of E_{n|k} and generalizes key divisibility theorems related to q-tangent numbers, expanding understanding of these combinatorial objects.

Findings

Established divisibility properties of the q-analog of E_{n|k}

Generalized theorems of Andrews and Gessel for q-tangent numbers

Extended classical Euler number results to a broader q-analog context

Abstract

The generalized Euler number E_{n|k} counts the number of permutations of {1,2,...,n} which have a descent in position m if and only if m is divisible by k. The classical Euler numbers are the special case when k=2. In this paper, we study divisibility properties of a q-analog of E_{n|k}. In particular, we generalize two theorems of Andrews and Gessel about factors of the q-tangent numbers.