Higher derivatives of the inverse tangent function and a summation formula involving binomial coefficients

Abstract

In 2017, O. Deiser and C. Lasser obtained an explicit formula for the $n$-th derivative of the inverse tangent function. We calculate this derivative by a different method based on Faà di Bruno's formula. Comparing the two results leads to the following identity for binomial coefficients: $$\sum _{i=m}^{[n/2]}\frac{(-1{)}^i}{4^i}\left(\genfrac{}{}{0ex}{}{i}{m}\right)\left(\genfrac{}{}{0ex}{}{n-i}{i}\right)=\frac{(-1{)}^m}{2^n}\left(\genfrac{}{}{0ex}{}{n+1}{2m+1}\right),$$ where $n,m\in {\mathbb{N}}_0$ and $m\le [n/2]$. As was pointed out to the author by C. Krattenthaler, this formula is a special case of Gauß's formula for the hypergeometric function ${}_2{F}_1$.

8 pages