On irrationality of Euler's constant and related asymptotic formulas

By defining $$I_n:={\int }_0^1{\int }_0^1\frac{(x(1-x)y(1-y){)}^n}{(1-xy)(-\log xy)}dxdy$$ Sondow (see [2]) proved that $$I_n=\left(\genfrac{}{}{0px}{}{2n}{n}\right)\gamma +L_n-A_n$$ We prove asymptotic formula for $L_n$ and $A_n$ as $n\to \mathrm{\infty }$, $$L_n=\left(\genfrac{}{}{0px}{}{2n}{n}\right)(\log \left(\frac{3n}{2}\right)+\mathcal{O}\text{\hspace{0.17em}⁣}\left(\frac{1}{n}\right))$$ and $$A_n\sim \frac{4^n}{\sqrt{\pi n}}(\gamma +\ln \frac{3}{2}+\ln n)$$ Using the sufficient condition for irrationality criteria of Euler’s constant due to Sondow, we prove that $\gamma$ is irrational.