Explicit estimates for the sum $\sum_{k=0}^{n} k! {n\choose k}^2 (-1)^{k}$

Abstract

We are interested in finding an explicit estimate to the binomial sum $Q_n(x)=\sum_{k=0}^{n} k! {n\choose k}^2 (-x)^{k}$ at $x=1$ for $n=0,1,2,\ldots$. Despite of its own interest the polynomial $Q_n(x)$ is important as the denominator in the Pad\'e 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 $n=0,1,2,\ldots$, which replaces the Fej\'er'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.