Evaluations of sums involving harmonic numbers and binomial coefficients. Issue 7 (3rd July 2019) - Wang, Weiping; Xu, Ce
ABSTRACT: In this paper, by the Faà di Bruno formula, we establish the decompositions of two general fractions involving the reciprocals of products of binomial coefficients. Using the decompositions, we discuss the evaluations of some Euler-type sums involving harmonic numbers and binomial coefficients, such as S π 1, q ( k ) = ∑ n = 1 ∞ H n ( π 1 ) n q ∏ i = 1 p n + k i k i, S π 1 q ( k ) = ∑ n = 1 ∞ n q H n ( π 1 ) ∏ i = 1 p n + k i k i, and some other forms. We present some explicit evaluations as examples and provide the Maple package to compute the sums S π 1, q ( k ) and S π 1 q ( k ) . It can be found that this work gives a unified approach to such sums and generalizes many known results in the literature.