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

In this paper, we define the deformed Euler $(s,t)$-numbers ${\mathrm{e}}_{s,t,u}$ Furthermore, we prove that ${\mathrm{e}}_{as,a^{2}t,u^{-1}}$ and ${\mathrm{e}}_{as,a^{2}t,u^{-1}}^{-1}$ are irrational numbers when $a,u\in \mathbb{Q}$ and $\mathrm{\mid }au\mathrm{\mid }>1$, 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.