Hypergeometric series and harmonic number identities

Abstract

The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (

Hypergeometric approach to Weideman’s conjecture

“…This technique has been deviced and explored recently in [4] and [8]. In particular, the present paper will focus our attention on a new proof of the identity (1) by twice differentiating the reformulated Dixon formula.…”

Generalizations of Two Summation Formulas for the Generalized Hypergeometric Function of Higher Order Due to Exton

“…Exton [2] obtained some interesting summation formulas. Among those things, we recall here the following two formulas written in slightly modified form:…”

“…In fact, we derive 22 (11 each) summation formulas for 4 ; 1 which are the special cases of Equation (1.4) in [2] and the corrected form of Equation (1.5) in [2], respectively. Note that the q+1 F q appearing on the righthand side of Equation (1.5) in the work of Exton [2] should be corrected as 2q+2 F 2q+1 .…”

“…For detailed discussion of this function, including the convergence of its series representation, see, for example, Exton [1], Slater [5], Rainville [4], or Srivastava and Choi [6]. Among other things in the theory and application of p F q , the summation formulas for p F q such as (1.4) and (1.5) have played vital roles (see, e.g., Shen [7], Choi and Srivastava [1], Chu and de Donno [2]). By considering the following two combinations…”

Infinite series identities involving quadratic and cubic harmonic numbers

“…There are several approaches to evaluate these sums. By applying derivative operator to the known binomial identities and terminating hypergeometric formulae, Chu and De Donno [9,11,12] established numerous summation formulae. Boyadzhiev [2] evaluated several finite sums by the Euler transformation.…”