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

TL;DR

This paper derives higher derivatives of the inverse tangent function using Faà di Bruno's formula and uncovers a binomial coefficient identity related to hypergeometric functions.

Contribution

It presents a novel derivation method for the derivatives and establishes a new binomial coefficient identity connected to hypergeometric functions.

Findings

Derived explicit formula for higher derivatives of arctangent

Established a new binomial coefficient summation identity

Connected the identity to Gauss's hypergeometric formula

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 Fa\`a di Bruno's formula. Comparing the two results leads to the following identity for binomial coefficients: $$\sum_{i=m}^{[n/2]}\frac{(-1)^i}{4^i}\binom{i}{m}\binom{n-i}{i}=\frac{(-1)^m}{2^n}\binom{n+1}{2m+1},$$ where $n,m\in \mathbb{N}_0$ and $m\leq [n/2]$. As was pointed out to the author by C. Krattenthaler, this formula is a special case of Gau\ss's formula for the hypergeometric function $_2F_1$.