1,208 Results Found

The Euler–Mascheroni constant $\gamma$, defined as the limiting difference between the harmonic numbers $H_n$ and $logn$, has long been studied and appears in diverse areas of number theory, analysis, and special functions. In this paper, we establis...

Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are immediate con...

Harmonic numbers appear, for example, in many expressions involving Riemann zeta functions. Here, among other things, we introduce and study discrete versions of those numbers, namely the discrete harmonic numbers. The aim of this paper is twofold. T...

The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.

Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: $H_n\sim \gamma +logn-{\sum }_{k=1}^{\infty }\frac{B_k}{k·n^k}$, where $B_k$ is the Bernoulli numbers. In this paper, we rewrite Ramanujan’s harmonic numbe...

In this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and...

By computing definite integrals, we shall examine binomial series of convergence rate $\pm 1/2$ and weighted by harmonic-like numbers. Several closed formulae in terms of the Riemann and Hurwitz zeta functions as well as logarithm and polylogarithm...

We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new.

By employing the coefficient extraction method, a class of binomial series involving harmonic numbers will be reviewed through three hypergeometric ${}_{2}F_1\left(y^2\right)$-series. Numerous closed-form generating functions for infinite series containing binomial coef...

Two general transformations for hypergeometric series are examined by means of the coefficient extraction method. Several interesting closed formulae are shown for infinite series containing harmonic numbers and binomial/multinomial coefficients. Amo...

Eight infinite series involving harmonic-like numbers are coherently and systematically reviewed. They are evaluated in closed form exclusively by integration together with calculus and complex analysis. In particular, a mysterious series $\mathfrak{W}$ is introd...

In this paper, the limit points of the sequence of arithmetic means ${\displaystyle \frac{1}{n}}{\sum }_{m=1}^n{\left\{H_m\right\}}^{\sigma }$ for $n=1,2,3,\dots$ are studied, where $H_m$ is the mth harmonic number with fractional part $\left\{H_m\right\}$ and $\sigma$ is a fixed positive constant. In particular, for $\mathrm{\&s}$...

We study the formulas for binomial sums of harmonic numbers of higher order ${\displaystyle \sum _{k=0}^n}{H}_k^{\left(r\right)}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){(1-q)}^k{q}^{n-k}=H_n^{\left(r\right)}-{\displaystyle \sum _{j=1}^n}{\mathcal{D}}_r(n,j){\displaystyle \frac{q^j}{j}}.$ Recently, Mneimneh proved that ${\mathcal{D}}_1(n,j)=1$. In this paper, we find several different expressions of ${\mathcal{D}}_r(n$...

The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, o...

In this work, we consider a problem from the field of power-aware scheduling in which a fleet of electric vehicles have to be charged in a minimum time. Each vehicle is equipped with a lithium-ion battery of a given capacity. The initial power used f...

In the present work, we define harmonic complex balancing numbers by considering well-known balancing numbers and inspiring harmonic numbers. Mainly, we investigate some of their basic characteristic properties such as the Binet formula and Cassini i...

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 presentation is the r...

A neutrosophic number (a + bI) is a significant mathematical tool to deal with indeterminate and incomplete information which exists generally in real-world problems, where a and bI denote the determinate component and indeterminate component, respec...

In this article, the up and down harmonically convex fuzzy-number-valued mapping which is a novel kind of harmonically convex fuzzy-number-valued mapping is introduced. In addition, it is highlighted that the new idea of up and down harmonically conv...

We propose the concept of up and down harmonically convex mapping for fuzzy-number-valued mapping as our main goal in this work. With the help of up and down harmonically fuzzy-number convexity and the fuzzy fractional integral operator, we also show...

Many fields of mathematics rely on convexity and nonconvexity, especially when studying optimization issues, where it stands out for a variety of practical aspects. Owing to the behavior of its definition, the idea of convexity also contributes signi...

In this paper, we provide different variants of the Hermite–Hadamard ($H\cdot H$) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically $s$-convex fuzzy-number-valued functions ($U\mathcal{D}\mathcal{H}$ $s$-...

Two classes of series involving differences of harmonic numbers and the binomial coefficients $C(3n,n)$ are evaluated in closed form. The classes under consideration are ${\sum }_{k=0}^{\infty }{\displaystyle \frac{H_{3k+1}-H_k}{(3k+1)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)}}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{m}}\right)z^k\mathrm{and}{\sum }_{k=0}^{\infty }{\displaystyle \frac{H_{2k}-H_k}{(3k+1)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)}}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{}}\right)$...

This study presents the characterization of the aerodynamic forces and moments acting on irregular particles of prescribed sphericity, generated through truncated spherical harmonic expansions and immersed in a uniform flow at intermediate Reynolds n...

Picture fuzzy sets (PFSs) can be used to handle real-life problems with uncertainty and vagueness more effectively than intuitionistic fuzzy sets (IFSs). In the process of information aggregation, many aggregation operators under PFSs are used by dif...

The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis func...

The Pell numbers, named after the English diplomat and mathematician John Pell, are studied by many authors. At this work, by inspiring the definition harmonic numbers, we define harmonic Pell numbers. Moreover, we construct one type of symmetric mat...

In this work, we establish some characteristics for a sequence, ${\mathcal{A}}_{\alpha }(n,k)$, including recurrence relations, generating function and inversion formula, etc. Based on the sequence, we derive, by means of the generating function approach, some transf...

We examine a useful hypergeometric transformation formula by means of the coefficient extraction method. A large class of “binomial/harmonic series” (of convergence ratio “$1/4$”) containing the cubic central binomial coefficien...

We develop a number of new identities for linear Euler harmonic number sums with multiple argument. Some new examples of even weight linear Euler harmonic sum identities are given.

We study inverse-binomial double series involving ordinary and alternating harmonic numbers. Using iterated integrals associated with admissible 3-posets, we establish twenty-four explicit evaluations in terms of multiple zeta values and alternating...

Power inverters are extensively employed in a wide range of applications such as VSD, PV, UPS, and Vehicle-to-Grid systems. Different control topologies are used in power electronic inverters. Of these, PWM and MLVSIs are implemented to minimize the...

In this paper, $\left(p,q\right)$-analogues of the generalized harmonic polynomials and the generalized hyperharmonic polynomials of order $\nu$ are defined. We found new sums of these numbers by using combinatorial operations. We also obtained the generating functi...

This paper aims to improve the energy autonomy and the mechanical performances of an on-board drive chain for robotics. The energy autonomy improvement is performed by reducing electrical losses in the inverter. Electrical losses are reduced by decre...

Binomial series involving harmonic polynomials are expressed in terms of parametric integrals. By evaluating these parametric integrals, we establish several remarkable closed formulae for the infinite series containing both central binomial coeffici...

This paper demonstrates several of Ostrowski-type inequalities for fuzzy number functions and investigates their connections with other inequalities. Specifically, employing the Aumann integral and the Kulisch–Miranker order, as well as the inc...

This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatoria...

We present a spectrally-dense phase-reference and its calibration for nonlinear vector network analyzers (NVNAs) using a step recovery diode (SRD) comb-generator with a multi-tone stimulus. Frequency selection for multi-tone stimulus based on prime n...

Two summation theorems concerning the $F_1^2(1/2)$-series due to Gauss and Bailey will be examined by employing the “coefficient extraction method”. Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will...

We develop a unified method for transforming and evaluating infinite series involving products of terms of second-order linear recurrences in the denominator. The approach is based on a discrete Abel-type summation formula (summation by parts), which...

In this paper, we express ${\sum }_{n,m\ge 1}{\displaystyle \frac{{\mathit{\epsilon }}_1^n{\mathit{\epsilon }}_2^m{M}_n^{\left(u\right)}{M}_m^{\left(v\right)}}{n^r{m}^s{(n+m)}^t}}$ as a linear combination of alternating multiple zeta values, where ${\mathit{\epsilon }}_i\in \{1,-1\}$ and $M_k^{\left(u\right)}\in \{H_k^{\left(u\right)},{\overline{H}}_k^{\left(u\right)}\}$, with $H_k^{\left(u\right)}$ and ${\overline{H}}_k^{\left(u\right)}$ being har...

By means of the coefficient extraction method, we examine a transformation of a classical hypergeometric series. Three classes of infinite series (of convergence rate “$1/4$”) with harmonic numbers in numerators and cubic central binomial c...

By applying the “coefficient extraction method” to the symmetric transformation of hypergeometric series due to Chu and Zhang (2014), an overview is presented systematically for a large class of infinite series of convergence rate “...

In this paper, we discuss various estimates to the right-hand (resp. left-hand) side of the Hermite–Hadamard inequality for functions whose absolute values of the second (resp. first) derivatives to positive real powers are log-convex. As an ap...

This study presents new fuzzy adaptations of Ostrowski’s integral inequalities through a novel class of convex fuzzy-valued mappings defined over a harmonic convex set, avoiding the use of the Sugeno integral. These innovative inequalities gene...

In a mixture of two kinds of identical bosons, there are two types of pairs: identical bosons’ pairs, of either species, and pairs of distinguishable bosons. In the present work, the fragmentation of pairs in a trapped mixture of Bose–Ein...

The purpose of this experimental study was to evaluate the range of particulate mass (PM) and particulate number (PN) results from gasoline direct injection (GDI) vehicles by using four test fuels with a range of particulate matter index (PMI) from 1...

Utilizing non-overlap windings has emerged as a favourable choice for minimizing electrical machine manufacturing costs, among other benefits. Nevertheless, it is widely acknowledged that these windings exhibit a notable level of harmonic contents in...

This study investigates the aerodynamic performance of a symmetric NACA 0018 airfoil under harmonic pitching motions at low Reynolds numbers, a regime characterized by the presence of laminar separation bubbles and their impact on aerodynamic forces....

A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we prese...