Reciprocal Binomial Coefficients Research Articles

Generalized hyperharmonic number sums with reciprocal binomial coefficients

Abstract In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.

Euler sums of generalized harmonic numbers and connected extensions

This paper presents the evaluation of the Euler sums of generalized hyper-harmonic numbers Hn(p,q) in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of the Riemann zeta values.

Euler-type sums involving multiple harmonic sums and binomial coefficients

Abstract In this paper, we mainly show that generalized Euler-type sums of multiple harmonic sums with reciprocal binomial coefficients can be expressed in terms of rational linear combinations of products of classical multiple zeta values (MZVs) and multiple harmonic star sums (MHSSs). Furthermore, applying the stuffle relations, we prove that the Euler-type sums involving products of generalized harmonic numbers and reciprocal binomial coefficients can be evaluated by MZVs and MHSSs.

On Evaluations of Euler-Type Sums of Hyperharmonic Numbers

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers \(h_{n}^{\left( r\right) }\) with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of hyperharmonic numbers to an arbitrary integer r. Moreover, we reach at explicit formulas for the shifted Euler-type sums of harmonic and hyperharmonic numbers. All the evaluations are provided in terms of the Riemann zeta values, harmonic numbers and linear Euler sums.

Generalized alternating hyperharmonic number sums with reciprocal binomial coefficients

In this paper, we show that generalized alternating hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.

Euler Related Binomial Sums

We develop new closed form representations of sums of reciprocal binomial coefficients. We also identify new integral and hypergeometric representation for the binomial-harmonic number sums.

Half-Integer Values of the Sums of Harmonic Numbers of Order Two

Half-integer values of the sums of harmonic numbers and reciprocal binomial coefficients are investigated. Closed-form representations and integral expressions are developed for the infinite series.

Quadratic and cubic harmonic number sums

This paper develops an approach to evaluation of Euler related sums. The approach is based on simple integral computations. By the approach, we can obtain some closed form representations of sums of quadratic and cubic harmonic numbers and reciprocal binomial coefficients. The given representations are new.

Some evaluation of harmonic number sums

ABSTRACTIn this paper, by using the method of partial fraction decomposition and integral representations of series, we establish some expressions of series involving harmonic numbers and binomial coefficients in terms of zeta values and harmonic numbers. Furthermore, we can obtain some closed form representations of sums of products of quadratic (or cubic) harmonic numbers and reciprocal binomial coefficients, and some explicit evaluations are given as applications. The given representations are new.

Harmonic numbers at half integer values

ABSTRACTHalf integer values of harmonic numbers and reciprocal binomial coefficients sums are investigated in this paper. Closed-form representations and integral expressions are developed for the infinite series.

Second order alternating harmonic number sums

We develop new closed form representations of sums of alternating harmonic numbers of order two and reciprocal binomial coefficients. Moreover we develop new integral representations in terms of harmonic numbers of order two.

New families of alternating harmonic number sums

We develop new closed form representations of sums of alternating harmonic numbers and reciprocal binomial coefficients.

Finite Mixed Sums wih Harmonic Terms

Abstract In the spirit of Euler sums we develop a set of identities for finite sums of products of harmonic numbers in higher order and reciprocal binomial coefficients. The new results complement some Euler sums of the type

Quadratic alternating harmonic number sums

We develop new closed form representations of sums of quadratic alternating harmonic numbers and reciprocal binomial coefficients.

A family of shifted harmonic sums

In this paper, we first develop a set of identities for Euler-type sums. We then investigate products of the shifted harmonic numbers and the reciprocal binomial coefficients. We briefly indicate relevant connections of the results presented here with those given by earlier authors. As by-products of our investigation, we derive several (presumably new) one-parameter and two-parameter summation formulas for the hypergeometric series 3F2 at argument 1.

SHIFTED HARMONIC SUMS OF ORDER TWO

We develop a set of identities for Euler type sums. In par- ticular we investigate products of shifted harmonic numbers of order two and reciprocal binomial coefficients. 1. Introduction and preliminaries In this paper we study the summation of the product shifted harmonic sums of order two and reciprocal binomial coefficients, of the form, ∞ X n=1 H (2) n− 1

One Class Series of Reciprocals of Non-Central Binomial Coefficients

Utilizing Gamma-Beta function, we can build one series involving reciprocal of non-central binomial coefficients, then We can structure several new series of reciprocals of non-central binomial coefficients by item splitting, these new created denominator of series contain 1 to 4 odd factors of binomial coefficients. As the result of splitting items, some identities of series of numbers values of reciprocals of binomial coefficients are given. The method of splitting terms offered in this paper is a new combinatorial an analysis way and elementary method to create new series.

One Class Series Involving Reciprocals of Binominal Coefficients

Using one known series, We can structure several new series of reciprocals of central binomial coefficients by splitting terms ,these new created denominator of series contain 1 to 4 factors of binomial coefficients. As the result of splitting terms, some identities of series of numbers values of reciprocals of binomial coefficients are given. The method of splitting terms offered in this paper is a new combinatorial an analysis way and elementary method to create new series.

Mixed binomial sum identities

We investigate sums of reciprocal binomial coefficients with quadratic harmonic numbers and harmonic numbers of order two. We create new analytical identities in closed form of Euler-related sums.

Euler-related sums

Purpose The purpose of this paper is to develop a set of identities for Euler type sums of products of harmonic numbers and reciprocal binomial coefficients.