Related papers: The Roman harmonic numbers revisited

Two classes of infinite series involving harmonic numbers and the binomial coefficient $C(3n,n)$ are evaluated in closed form using integrals. Several remarkable integral values and difficult series identities are stated as special cases of…

We propose a relation between values of the Riemann zeta function $\zeta$ and a family of integrals. This results in an integral representation for $\zeta(2p)$, where $p$ is a positive integer, and an expression of $\zeta(2p+1)$ involving…

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ \zeta_{H^{\left( p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left( p,q\right) }}{n^{r}}%…

In this paper, we study a Dirichlet series generated by powers of harmonic numbers. As an application of these functions, we derive certain series involving harmonic numbers. We also study the analytic properties of these Dirichlet series…

In this paper, we first introduce certain forms of extended incomplete Pochhammer symbols which are then used to define families of extended incomplete generalized hypergeometric functions. For these functions, we investigate various…

We evaluate some new three parameter families of finite reciprocal sums involving Horadam numbers. We will also be able to state the results for the infinite sums. Some Fibonacci and Lucas sums will be presented as examples.

We first present some identities involving the Pochhammer symbol (rising factorial). We also recall and present some new properties of the Jacobi polynomials. We use them to expand a general hypergeometric function in an orthogonal series…

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 evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these…

Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the…

A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving…

In this paper, we take advantage of the Mellin type derivative to produce some new families of polynomials whose coefficients involve r-Lah numbers. One of these polynomials leads to rediscover many of the identities of r-Lah numbers. We…

The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).

The denominators $d_n$ of the harmonic number $1+\frac12+\frac13+\cdots+\frac1n$ do not increase monotonically with~$n$. It is conjectured that $d_n=D_n={\rm LCM}(1,2,\ldots,n)$ infinitely often. For an odd prime $p$, the set…

Following the Mellin and inverse Mellin transform techniques presented in our paper arXiv:1606.02150 (NT), we have established close forms of Laurent series expansions of products of bi- and trigamma functions /psi(z)*/psi(-z) and…

This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…

A new musical scale devised by the author, based on natural logarithms, is described. Most of the logarithmic pitches bear no correspondence to the twelve tones of the ancient tuning system attributed to Pythagoras, based on ratios of whole…

For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum_{0<k\le n}\frac1{k^m}\ \ (n=0,1,2,\ldots).$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For…

Harmonic numbers $H_k=\sum_{0<j\le k}1/j (k=0,1,2,...)$ arise naturally in many fields of mathematics. In this paper we initiate the study of congruences involving both harmonic numbers and Lucas sequences. One of our three theorems is as…

We evaluate integrals of certain polynomials over spheres and balls in real or complex spaces. We also promote the use of the Pochhammer symbol which gives the values of our integrals in compact forms.