Some identities involving rational sums

Abstract

In this note a procedure to deduce new identities involving elementary rational sums is presented and applying this technique some sums including binomial coefficients and harmonic numbers are obtained.

Some combinatorial series identities and rational sums

“…Remark 5 Theorem 1 of Díaz-Barrero et al [2] would follow from the assertion (27) of Corollary 1 when we set λ = −n − 1 (n ∈ N) and apply the combinatorial identity (29). Alternatively, by merely putting μ = 0 in the assertion (31) of Corollary 2, we arrive at the combinatorial series identity (5) Remark 6 With a view to deducing the combinatorial series identity (6) asserted by Theorem 2, we simply set λ = −n − 1 (n ∈ N) in the assertion (28) of Corollary 1 and make use the combinatorial identity (29) or (alternatively) μ = 0 in the assertion (32) of Corollary 2.…”

“…Díaz-Barrero et al [2] presented a procedure to derive generalizations and extensions of the identity (4) given by Theorems 1 and 2.…”

“…At least two subsequent works [6,10] have been motivated by the aforementioned investigation by Díaz-Barrero et al [2]. While Sofo [10] used the method of integral representations in order to recapture Theorems 1 and 2 and to highlight closed-form evaluations of other combinatorial sums, the work of Prodinger [6] provided an alternative approach to such series identities as those given by Theorems 1 and 2, which is based upon the principles of inverse pairs and partial-fraction decomposition (see also the references cited in each of the earlier works [6,10]).…”

Combinatorial identities from contour integrals of rational functions

“…In addition, Eq. (4.5) contains many known identities such as the identity of Barrero-Báguena-Popescu [6], the identity of Guo and Zhang [14], the identities of Prodinger [21,22], the identity of Mansour-Shattuck-Song [17], and the identity of Chu and Yan [3,4]. It also contains several identities involving harmonic numbers and q-harmonic numbers [16,25,26].…”

On a General $q$-Identity

“…See Theorem 2.2 in [13]. In the paper [5], Díaz-Barrero et al obtained two identities involving rational sums:…”