Higher derivatives and some more identities involving generalized harmonic numbers and Bernoulli polynomials and Bernoulli numbers

Abstract

In the present work, we focus on the higher derivatives of polynomials and certain rational fractions expressed in terms ofthe well-known complete Bell polynomials. As consequences, we obtain explicit formulas of the higher derivativesof the binomial coefficient and its reciprocal. Our results represent a unified generalization ofmany previously presented works and provide a natural way to establish several new algebraic identities.Furthermore, we provide various interesting combinatorial identities involving the harmonic numbers, the generalized harmonic numbers, the Bernoulli numbers, and the Bernoulli polynomials.