q-Euler and Genocchi numbers

TL;DR

This paper introduces a new construction of $q$-Euler numbers, defines $q$-Genocchi numbers based on them, and explores their relationships, expanding the understanding of $q$-analogues of classical number sequences.

Contribution

It presents a novel construction of $q$-Euler numbers differing from Carlitz's, and develops $q$-Genocchi numbers with their interrelations.

Findings

New $q$-Euler number construction

Definition of $q$-Genocchi numbers

Relations between $q$-Euler and $q$-Genocchi numbers

Abstract

Carlitz has introduced an interesting $q$-analogue of Frobenius-Euler numbers in [4]. He has indicated a corresponding Stadudt-Clausen theorem and also some interesting congruence properties of the $q$-Euler numbers. In this paper we give another construction of $q$-Euler numbers, which are different than his $q$-Euler numbers. By using our $q$-Euler numbers, we define the $q$-analogue of Genocchi numbers and investigate the relations between $q$-Euler numbers and $q$-analogs of Genocchi numbers.