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In this paper a novel operational matrix of derivatives of certain basis of Legendre polynomials is established. We show that this matrix is expressed in terms of the harmonic numbers. Moreover, it is utilized along with the collocation method for handling initial value problems of any order. The convergence and the error analysis of the proposed expansion are carefully investigated. Numerical examples are exhibited to confirm the reliability and the high efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

It is an open question asked by Mezö that there is no hyperharmonic integer except 1. So far it has been proved that all hyperharmonic numbers are not integers up to order r = 25 . In this paper, we extend the current results for large orders. Our method will be based on three different approaches, namely analytic, combinatorial and algebraic. From analytic point of view, by exploiting primes in short intervals we prove that almost all hyperharmonic numbers are not integers. Then using combinatorial techniques, we show that if n is even or a prime power, or r is odd then the corresponding hyperharmonic number is not integer. Finally as algebraic methods, we relate the integerness property of hyperharmonic numbers with solutions of some polynomials in finite fields. [ABSTRACT FROM AUTHOR]

We study the well-known lacunary sums of binomial coefficients considered, most notably, by Christian Ramus, and their connection to a special kind of harmonic number associated with the first case of Fermat's Last Theorem. For one case of Ramus's famous identity we obtain a variation in which some of the parameters are replaced by square matrices of arbitrary dimension. [ABSTRACT FROM AUTHOR]

We employ Stirling numbers of the second kind to prove a relation of Riordan involving harmonic numbers. [ABSTRACT FROM AUTHOR]

This paper analyzes the solution of fourth-order linear and nonlinear two point boundary value problems. The suggested method is quite innovative and it is completely different from all previous methods used for solving such kind of boundary value problems. The method is based on employing an elegant operational matrix of derivatives expressed in terms of the well-known harmonic numbers. Two algorithms are presented and implemented for obtaining new approximate solutions of linear and nonlinear fourth-order boundary value problems. The two algorithms rely on employing the new introduced operational matrix for reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. For this purpose, the two spectral methods namely, Petrov-Galerkin and collocation methods are applied. Some illustrative examples are considered aiming to ascertain the wide applicability, validity, and efficiency of the two proposed algorithms. The obtained numerical results are satisfactory and the approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature.

Let n ∈ N = { 1 , 2 , 3 , ... } and 0 ≤ p ≤ 1. We show that ∑ k = 0 n H k ( n k ) p k q n − k = ∑ i = 1 n 1 − q i i where H k = ∑ i = 1 k 1 / i is the k th harmonic number and q = 1 − p. This generalizes a previously known identity for the special case of p = 1 / 2. We derive several forms related to this sum and show how they can be useful in the probabilistic analysis of some simple variants of existing problems. [ABSTRACT FROM AUTHOR]

The applications of spreadsheets to mathematics allow learners to analyze mathematical ideas in ways never before possible. In this paper harmonic numbers are defined, calculated, approximated, and applied. The concepts and techniques of recursion, approximation and combinatorial methods are illustrated. This paper also shows some properties of the harmonic series including divergence and its part in the determination of harmonic numbers. Analyzing the ideas of harmonic numbers presents to learners that one concept has many connections to others. Mathematical concepts and relationships between harmonic numbers, harmonic series, the Euler-Mascheroni constant and the harmonic mean are explored. Finding expressions in closed-form is not always possible and technology provides the capability to approximate the harmonic numbers. Exploration and experimentation are discussed. With spreadsheets, complex calculations can be simplified, intuition can be developed and applications can be studied.

By means of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established. [ABSTRACT FROM PUBLISHER]

For any prime number p , let J p be the set of positive integers n such that p divides the numerator of the n -th harmonic number H n . An old conjecture of Eswarathasan and Levine states that J p is finite. We prove that for x ≥ 1 the number of integers in J p ∩ [ 1 , x ] is less than 129 p 2 / 3 x 0.765 . In particular, J p has asymptotic density zero. Furthermore, we show that there exists a subset S p of the positive integers, with logarithmic density greater than 0.273, and such that for any n ∈ S p the p -adic valuation of H n is equal to − ⌊ log p ⁡ n ⌋ . [ABSTRACT FROM AUTHOR]

In this paper, using some combinatorial identities, we present new congruences involving central binomial coefficients and harmonic, Catalan, and Fibonacci numbers. For example, for an odd prime p, we have ... where {Fn} is the Fibonacci sequence and {Qn} is the Pell-Lucas sequence. [ABSTRACT FROM AUTHOR]

An interplay between the sum of certain series related to harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits us to find sharp inequalities bounding these trigonometric sums. In particular, this answers positively an open problem of Chen (Excursions in Classical Analysis, ). [ABSTRACT FROM AUTHOR]

We prove that if λ ⩽ 1=2, then ζ(3) = Σ n=1 1 (1 - λ)n+1 Σ k=1 (n k) (-λ)n-kδk with δk = Hk k2 1 k л 2 6 H(2) k; where Hk and H(2)k denote the harmonic numbers and the generalized harmonic numbers of order 2, respectively. [ABSTRACT FROM AUTHOR]

The harmonic numbers play important roles in mathematics. Let be a prime. With the help of some combinatorial identities, we establish the following two new congruences: and where denotes the Bernoulli polynomial of degree . As an application, we determine and modulo , where with . [ABSTRACT FROM AUTHOR]

In this paper, we investigate certain asymptotic series for the tail of the Riemann Zeta function used by M. D. Hirschhorn to prove an asymptotic expansion of Ramanujan for the $$n$$ th Harmonic number. We give a general form of these series with an explicit formula for its coefficients and with a precise error term. [ABSTRACT FROM AUTHOR]

The paper is about calculating multiple fractional part integrals of the form ...where {x} denotes the fractional part of x and k,m and p are nonnegative integers. We show that these integrals can be expressed as series involving products of Riemann zeta function values and some binomial coefficients. We obtain, as particular cases of our results, new integral representations of Euler's constant as double and triple symmetric fractional part integrals. [ABSTRACT FROM AUTHOR]

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) γ m are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in π − 2 with rational coefficients and converges slightly better than Euler's series ∑ n − 2 . The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A , the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them. [ABSTRACT FROM AUTHOR]

In the first part of this paper we obtained several bi-unitary harmonic numbers which are higher than 109, using the Mersenne prime numbers. In this paper we investigate bi-unitary harmonic numbers of some particular forms: 2k · n, pqt², p²q²t, with different primes p, q, t and a squarefree integer n. [ABSTRACT FROM AUTHOR]

Text The Arakawa–Kaneko zeta functions interpolate the poly-Bernoulli polynomials at the negative integers, while their values at the positive integers are connected to multiple zeta values and harmonic number sums. Here we construct p -adic analogues of these functions and show that these complex formulas have good p -adic counterparts. The method of construction, summing a series of forward differences of p -adic power functions, is a p -adic adaptation of an everywhere-convergent series for the Hurwitz zeta function due to Hasse. We also apply this method to construct a p -adic Lerch transcendent. Expressions for zeta values and related constants as harmonic number series which converge in both complex and p -adic senses are a prominent feature of this approach. Video For a video summary of this paper, please visit http://youtu.be/Mqd6W0TW9D0 . [ABSTRACT FROM AUTHOR]

In this paper, we provide a recurrence relation for determining the coefficients of Ramanujan's asymptotic expansion for the $$n$$ th harmonic number. [ABSTRACT FROM AUTHOR]

In terms of the derivative operator and a binomial sum from the telescoping method, we establish a family of summation formulae involving harmonic numbers. Meanwhile, a general recurrence relation is also deduced in accordance with Abel's lemma on summation by parts. [ABSTRACT FROM AUTHOR]

In [11], J. Sándor presented in a table all 211 bi-unitary harmonic num- bers up to 109. In the present paper we obtain several bi-unitary harmonic numbers which are higher than 109, using the Mersenne prime numbers. We also investigate bi-unitary harmonic numbers of some particular forms. Thus, we extend the study done by J. Sándor. [ABSTRACT FROM AUTHOR]

The asymptotic expansion of digamma function is a starting point for the derivation of approximants for harmonic sums or Euler–Mascheroni constant. It is usual to derive such approximations as values of logarithmic function, which leads to the expansion of the exponentials of digamma function. In this paper the asymptotic expansion of the function exp ( p ψ ( x + t ) ) is derived and analyzed in details, especially for integer values of parameter p . The behavior for integer values of p is proved and as a consequence a new identity for Bernoulli polynomials. The obtained formulas are used to improve know inequalities for Euler’s constant and harmonic numbers. [ABSTRACT FROM AUTHOR]

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 $$\;_3F_2\;$$ at argument 1. [ABSTRACT FROM AUTHOR]

This paper discusses generalization of some logarithmic integrals appeared in Adamchik, Medina and Moll's works. [ABSTRACT FROM AUTHOR]

In this note, we deal with a generalization of the harmonic numbers proposed by Conway and Guy in their Book of Numbers, namely, the so-called hyperharmonic numbers. Our main aim is to prove, by mathematical induction, the formula defining the hyperharmonic numbers in terms of ordinary harmonic numbers. Moreover, we calculate the hyperharmonic numbers as the derivative of a certain binomial coefficient. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend and refine the Ramanujan–Lodge harmonic number expansion into negative powers of a triangular number. We construct a faster asymptotic series and some new sharp inequalities for the harmonic numbers. [ABSTRACT FROM AUTHOR]

The paper is about calculating the following class of quadratic series:wherek≥2 is an integer. We show thatSkcan be expressed as a rational linear combination of zeta function values, binomial coefficients and products of two zeta function values. [ABSTRACT FROM AUTHOR]

We develop families of generalized hypergeometric function identities at argument±1 in terms of generalized harmonic numbers and Hurwitz zeta functions. [ABSTRACT FROM AUTHOR]

In this paper, we employ an expression of Boyadzhiev to give elementary proofs of the identities for harmonic numbers obtained by Paule- Schneider via the Mathematica package Sigma. [ABSTRACT FROM AUTHOR]

A generalized theorem related to harmonic sums is investigated from which we are able to represent an infinite number of sums of harmonic numbers in both multiple integral and closed forms. The new theorem generalizes known existing results in the published literature. [ABSTRACT FROM AUTHOR]

The multiple gamma function Γn(z), defined by a recurrence-functional equation as a generalization of the Euler gamma function, is used in many applications of pure and applied mathematics, and theoretical physics. The theory of the multiple gamma function has been related to certain spectral functions in mathematical physics, to the study of functional determinants of Laplacians of then-sphere, to HeckeL-functions, to the Selberg zeta function, and to the random matrix theory. There is a wide class of definite integrals and infinite sums appearing in statistical physics (the Potts model) and the lattice theory, which can be computed by means of the Γn(z) function. This paper presents new integral representations for the multiple gamma function and other mathematical functions and constants. [ABSTRACT FROM AUTHOR]

In the paper, two sharp inequalities for bounding the psi function ψ and the harmonic numbers H n are established respectively, some known results are improved and some remarks are given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we derive q-analogues of the multiparameter non-central Stirling numbers of the first and second kind, introduced by El-Desouky. Moreover, recurrence relations, explicit formulas and a connection between these numbers and generalized q-harmonic numbers are obtained. Furthermore, some important special cases and new combinatorial identities are given. Finally, algorithms of these numbers and matrix representation using Maple are derived. [Copyright &y& Elsevier]

This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for high-precision computation of the Barnes gamma function and Glaisher's constant are also discussed. [ABSTRACT FROM AUTHOR]