Related papers: On Two Types of Harmonic Number Identities
We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate…
The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…
The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate hyperharmonic numbers, hyperharmonic numbers and degenerate harmonic numbers. In particular, we derive an explicit expression…
We study two new classes of sums with inverse binomial coefficients and harmonic numbers. In addition we establish recursive solutions to the following power sums \begin{equation*} U_d(n) = \sum_{k=1}^n \frac{2^{2k}}{\binom{2k}{k}} \cdot…
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
In this work we prove a new combinatorial identity and applying it we establish many finite harmonic sum identities. Among many others, we prove that \begin{equation*}…
We present several types of ordinary generating functions involving central binomial coefficients, harmonic numbers, and odd harmonic numbers. Our results complement those of Boyadzhiev from 2012 and Chen from 2016. Based on these…
In this paper, we obtain formulas for the number of representations of positive integers as sums of arbitrarily many squares (and other polygonal numbers) with a certain natural weighting. The resulting weighted sums give Fourier…
The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The…
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as…
We present several sequences of Euler sums involving odd harmonic numbers. The calculational technique is based on proper two-valued integer functions, which allow to compute these sequences explicitly in terms of zeta values only.
The harmonic numbers are those $H_n=\sum_{0<k\le n}\frac1k\ (n=0,1,2,\ldots)$. In this paper we confirm over ten conjectural series identities with summands involving the binomial coefficient $\binom{4k}k$ and harmonic numbers. For example,…
We construct the generating function for products of inverse central binomial coefficients with harmonic numbers.
Some years ago, the harmonic polynomial was introduced in order to understand better the harmonic topological index; for instance, it allows to obtain bounds of the harmonic index of the main products of graphs. Here, we obtain several…
We introduce notions of bi-unitary, bi*-unitary and bi**-unitary harmonic numbers, along with their preliminary study.
We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.
We show that the compositions of positive integers may be interpreted in terms of powers of some power series, over arbitrary commutative ring. As consequences, several closed formulas for the compositions as well as for the generalized…
We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random…
Explicit evaluations of the Tornheim-like double series in the form \[ \sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm\left( n+m+s \right)},\ s\in \mathbb{N\cup } \left\{ 0 \right\} \] and their extensions are given. Furthermore, series of the type…
A recent paper of A. Sofo proves some results about sums of products of quadratic alternating harmonic numbers and reciprocal binomial coefficients. In this paper, we extend his result to cubic alternating harmonic number sums and develop…