Hyperharmonic Numbers Research Articles

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.

Euler sums of generalized alternating hyperharmonic numbers

We define the notion of the generalized alternating hyperharmonic numbers, and show that Euler sums of the generalized alternating hyperharmonic numbers can be expressed in terms of linear combinations of classical (alternating) Euler sums.

Sums of powers of integers and hyperharmonic numbers

In this paper, we obtain a new formula for the sums of k-th powers of the first n positive integers, Sk(n), that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Furthermore, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for Sk(n) to negative values of n.

Summation formulas of q-hyperharmonic numbers

In this paper, several weighted summation formulas of q-hyperharmonic numbers are derived. As special cases, several formulas of hyperharmonic numbers of type \(\sum _{\ell =1}^{n} {\ell }^{p} H_{\ell }^{(r)}\) and \(\sum _{\ell =0}^{n} {\ell }^{p} H_{n-\ell }^{(r)}\) are obtained.

Some applications on $q$-analog of the generalized hyperharmonic numbers of order $r,$ $ H_{n}^{r}(\alpha )$

In this paper, we define $q$-analog of the generalized harmonic numbers $H_{n}(\alpha )$ and the generalized hyperharmonic numbers of order $r,$ $H_{n}^{r}(\alpha ),$ and obtain some sums involving these numbers. Finally, we examine new applications of an $n\times n$ matrix $A_{n}=\left[ a_{i,j}\right] $ with the terms $a_{i,j}=H_{i}^{r}(j,q).$

Harmonic number identities via polynomials with r-Lah coefficients

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 show that some of these polynomials and hyperharmonic numbers are closely related. Taking into account of these connections, we reach several identities for harmonic and hyperharmonic numbers.

Euler sums and non-integerness of harmonic type sums

We show that Euler sums of generalized hyperharmonic numbers can be evaluated in terms of Euler sums of generalized harmonic numbers and special values of the Riemann zeta function. Then we focus on the non-integerness of generalized hyperharmonic numbers. We prove that almost all generalized hyperharmonic numbers are not integers and our error term is sharp and the best possible. Finally, we analyze generalized hyperharmonic numbers in terms of topology and relate this to non-integerness.

Some Results on Harmonic Type Sums

In this study, we consider the summatory function of convolutions of the Möbius function with harmonic numbers, and we show that these summatory functions are linked to the distribution of prime numbers. In particular, we give infinitely many asymptotics which are consequences of the Riemann hypothesis. We also give quantitative estimate for the moment function which counts non-integer hyperharmonic numbers. Then, we obtain the asymptotic behaviour of hyperharmonics.

Applications of derivative and difference operators on some sequences

In this study, depending on the upper and the lower indices of the hyperharmonic number h(r), nonlinear recurrence relations are obtained. It is shown that generalized harmonic numbers and hyperharmonic numbers can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hyperharmonic numbers are given. Negative-ordered hyperharmonic numbers are defined and their alternative representations are given.

Harmonic numbers associated with inversion numbers in terms of determinants

It has been known that some numbers, including Bernoulli, Cauchy, and Euler numbers, have such corresponding numbers in terms of determinants of Hessenberg matrices. There exist inversion relations between the original numbers and the corresponding numbers. In this paper, we introduce the numbers related to harmonic numbers in determinants. We also give several of their arithmetical and/or combinatorial properties and applications. These concepts can be generalized in the case of hyperharmonic numbers.

Identities between harmonic, hyperharmonic and Daehee numbers

In this paper, we present some identities relating the hyperharmonic, the Daehee and the derangement numbers, and we derive some nonlinear differential equations from the generating function of a hyperharmonic number. In addition, we use this differential equation to obtain some identities in which the hyperharmonic numbers and the Daehee numbers are involved.

On the matrices with the generalized hyperharmonic numbers of order r

In this paper, we define two [Formula: see text] matrices [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] are a generalized hyperharmonic numbers of order [Formula: see text]. We give some new factorizations and determinants of the matrices [Formula: see text] and [Formula: see text].

Computation and theory of Euler sums of generalized hyperharmonic numbers

Recently, Dil and Boyadzhiev [10] proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence ({0}r,1). In this paper, we show that the sums of multiple harmonic numbers whose indices are the sequence ({0}r,1;{1}k−1) can be expressed in terms of (multiple) zeta values, (multiple) harmonic numbers, and Stirling numbers of the first kind, and give an explicit formula.

Divisibility properties of hyperharmonic numbers

We extend Wolstenholme’s theorem to hyperharmonic numbers. Then, we obtain infinitely many congruence classes for hyperharmonic numbers using combinatorial methods. In particular, we show that the numerator of any hyperharmonic number in its reduced fractional form is odd. Then we give quantitative estimates for the number of pairs (n, r) lying in a rectangle where the corresponding hyperharmonic number $${ h_n^{(r)} }$$ is divisible by a given prime number p. We also provide p-adic value lower bounds for certain hyperharmonic numbers. It is an open problem that given a prime number p, there are only finitely many harmonic numbers h n which are divisible by p. We show that if we go to the higher levels r ≥ 2, there are infinitely many hyperharmonic numbers $${ h_n^{(r)} }$$ which are divisible by p. We also prove a finiteness result which is effective.

Structural properties of Dirichlet series with harmonic coefficients

An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers.

Evaluation of Euler-like sums via Hurwitz zeta values

In this paper we collect two generalizations of harmonic numbers (namelygeneralized harmonic numbers and hyperharmonic numbers) under one roof.Recursion relations, closed-form evaluations, and generating functions of thisunified extension are obtained. In light of this notion we evaluate someparticular values of Euler sums in terms of odd zeta values. We alsoconsider the noninteger property and some arithmetical aspects of this unifiedextension.

Almost all hyperharmonic numbers are not integers

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.

On shifted Mascheroni series and hyperharmonic numbers

TextIn this article, we study the nature of the forward shifted series σr=∑n>r|bn|n−r where r is a positive integer and bn are Bernoulli numbers of the second kind, expressing them in terms of the derivatives ζ′(−k) of zeta at the negative integers and Euler's constant γ. These expressions may be inverted to produce new series expansions for the quotient ζ(2k+1)/ζ(2k). Motivated by a theoretical interpretation of these series in terms of Ramanujan summation, we give an explicit formula for the Ramanujan sum of hyperharmonic numbers as an application of our results. VideoFor a video summary of this paper, please visit https://youtu.be/uyLmgDh9JVs.

Several explicit formulae of Cauchy polynomials in terms of $${r}$$ r -Stirling numbers

The integer values of Cauchy polynomials are expressed in terms of $${r}$$ -Stirling numbers of the first kind. Several relations between the integral values of Bernoulli polynomials and those of Cauchy polynomials are obtained in terms of $${r}$$ -Stirling numbers of both kinds. Also, we find a relation between the Cauchy polynomials and hyperharmonic numbers.

On the norms of r-circulant matrices with the hyperharmonic numbers

On the norms of r-circulant matrices with the hyperharmonic numbers