Harmonic number identities via the Newton–Andrews method

Abstract

By means of the Bell polynomials, we establish explicit expressions of the higher-order derivatives of the binomial coefficient $\binom{x+n}{m}$ and its reciprocal $\binom{x+n}{m}^{-1}$ , and extend the application field of the Newton–Andrews method. As examples, we apply the results to the Chu–Vandermonde–Gauss formula and the Dougall–Dixon theorem and obtain a series of harmonic number identities. This paper generalizes some works presented before and provides a way to establish infinite harmonic number identities.