Explicit estimates for the sum ${\sum }_{k=0}^{n}k!{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2(-1{)}^k$

Abstract

We are interested in finding an explicit estimate to the binomial sum $Q_n(x)={\sum }_{k=0}^{n}k!{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2(-x{)}^k$ at $x=1$ for $n=0,1,2,\dots$. Despite of its own interest the polynomial $Q_n(x)$ is important as the denominator in the Padé identity of the Euler's factorial series $E(x)={\sum }_{k=0}^{\infty }k!x^k$ as well as its close connection to a classical Laguerre polynomial $L_n(x)=\frac{1}{n!}{e}^x{\left(\frac{d}{dx}\right)}^n(e^{-x}{x}^n)$. Our main result is the explicit bound $$\left|L_n(1)-\sqrt{\frac{e}{\pi }}\cdot \frac{\cos (2\sqrt{n}-\frac{\pi }{4})}{n^{1/4}}+\frac{17}{48}\sqrt{\frac{e}{\pi }}\frac{\sin (2\sqrt{n}-\frac{\pi }{4})}{n^{3/4}}\right|<\frac{0.51}{n}$$ for all $n=0,1,2,\dots$, which replaces the Fejér's asymptotic formula from 1909. As a corollary of this, one also gets a new proof for the bound $|Q_n(1)|\le n!$, and even more.