Irrationality of the Deformed Euler Numbers ${\rm e}_{s,t,u}$

TL;DR

This paper introduces deformed Euler numbers and proves that certain transformations of these numbers are irrational, contributing to the understanding of their arithmetic nature.

Contribution

It defines the deformed Euler $(s,t)$-numbers and establishes their irrationality under specific parameter conditions, a novel step in studying their properties.

Findings

${ m e}_{as,a^2t,u^{-1}}$ is irrational for certain parameters.

The inverse ${ m e}_{as,a^2t,u^{-1}}^{-1}$ is also irrational under the same conditions.

Provides an infinite family of irrational numbers related to deformed Euler numbers.

Abstract

In this paper, we define the deformed Euler $(s,t)$-numbers ${\rm e}_{s,t,u}$ Furthermore, we prove that ${\rm e}_{as,a^2t,u^{-1}}$ and ${\rm e}_{as,a^2t,u^{-1}}^{-1}$ are irrational numbers when $a,u\in\mathbb{Q}$ and $\vert au\vert>1$, thus providing a countable infinite family of irrational numbers. This is the first step in a program to study the irrationality of $(s,t)$-analog of known numbers.