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The present manuscript deals with some certain finite sums and identities pertaining to some special numbers. Using generating functions methods, some relations and identities involving the Apostol type Euler and combinatorial numbers, and also the Fubini type numbers and polynomials, are given. Then, by using some certain classes of special finite sums involving the following rational sum which is defined by Simsek (2021b): y(r,ϑ)=∑_(b=0)^r▒〖(-1)^r/((1+b) ϑ^(b+1) 〖(ϑ-1)〗^(r-b+1) ),〗many new certain finite sums and formulas related to the Leibnitz, Harmonic, Changhee, and Daehee numbers are obtained. Moreover, some applications of these results are presented.

Introduction/purpose: Some sums of the polylogarithmic function associated with harmonic numbers are established. Methods: The approach is based on using the summation methods. Results: This paper generalizes the results of the zeta function series associated with the harmonic numbers. Conclusions: Various interesting series as the consequence of the generalization are obtained.

Yüksek Lisans Tezi, Bu tezde, ilk olarak harmonik Horadam sayıları tanımlanmıştır. Daha sonra, fark operatör yöntemi kullanılarak harmonik Horadam sayılarını içeren toplam formülleri elde edilmiştir. Ayrıca, elemanları harmonik Horadam sayıları olan circulant matrisler tanımlanmış ve bu matrislerin Euclidean, spektral, satır(sütun) normları elde edilmiştir., In this thesis, firstly the harmonic Horadam numbers are defined. Afterwards, sum formulas including harmonic Horadam numbers are obtained by using the difference operator method. In addition, circulant matrices with harmonic Horadam numbers are defined and Euclidean, spectral, row (column) norms of these matrices are obtained.

This paper introduces a generalization of the alternating harmonic series, expresses the sum in two closed forms, and examines the relationship between these sums and the harmonic numbers.

Recently the first author proved a congruence proposed in 2006 by Adamchuk: [Formula: see text] for any prime [Formula: see text]. In this paper, we provide more examples (with proofs) of congruences of the same kind [Formula: see text] where [Formula: see text] is a prime such that [Formula: see text], [Formula: see text] is a fraction in [Formula: see text] and [Formula: see text] is a [Formula: see text]-adic integer. The key ingredients are the [Formula: see text]-adic Gamma function [Formula: see text] and a special class of computer-discovered hypergeometric identities.

In the paper, we obtain asymptotic expansion of the finite sum of some sequences $ S_{n} = \sum_{k = 1}^{n}\left(n^{2}+k\right) ^{-1} $ by using the Euler's standard one of the harmonic numbers.

In this paper, we discuss sums of powers of the positive integers and compute both the exponential and ordinary generating functions for these sums. We express these generating functions in terms of exponential and geometric polynomials and also show their connection to other interesting series. In particular, we show their connection to an interesting problem of Ovidiu Furdui., 10 pages

We derive some series related to the polylogarithmic function, and we also give a new proof to the existing series. Our approach is based on using the summation and integral representation methods. We obtain various interesting series as a consequence.

In this paper, we derive new identities which are related to some special numbers and generalized harmonic numbers H_n (α) by using the argument of the generating function given in [3] and comparing the coefficients of the generating functions. Also considering q -numbers involving q -Changhee numbers Chnq and q-Daehee numbers Dnq, some sums are given. For example, for any positive integer n and any positive real number q > 1, whenα= q/(q-1), we have the relationship between generalized harmonic numbers and q -Daehee numbers

In this paper, the authors investigate two special families of series involving the reciprocal central binomial coefficients and Lucas numbers. Connections with several familiar sums representing the integer-valued Riemann zeta function are also pointed out.

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 combinatorial counterpart of the n-th prime number is derived together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π(x) are found.

Özet: Bu çalışmada$H_{z}^{(w)}=\frac{\left( z\right) _{w}}{z\Gamma\left( w\right) }\left( \Psi\left( z+w\right) -\Psi\left( w\right) \right)$where $\text{ \ \ }w\text{, }z+w\in\mathbb{C}\backslash\left( \mathbb{Z}^{-}\cup\left\{ 0\right\} \right).$eşitliği ile tanımlanan Hiperharmonik fonksiyonun bazı özellikleri araştırılmıştır. Bu tanımdan faydalanarak karmaşık indeksli harmonik sayılar tanıtılmış ve bu sayıların bazı serileri verilmiştir. Ayrıca rasyonel indeksli harmonik sayıların hesaplanması için formüller elde edilmiştir. $H_{z}^{(w)}$ fonksiyonunun türevlerinin daha kolay hesaplanabilmesi için, mevcut gösterim yeniden düzenlenmi¸stir. Bu yeni gösterim yardımıyla Hiperharmonik fonksiyonun yüksek mertebeli türevleri daha kolay hesaplanabilmektedir. Bunların yanı sıra, Hiperharmonik fonksiyonun özel bazı fonksiyonların birleşimi biçiminde ifade edilebildiği gerçeğinden hareketle, bazı özellikleri ve bağlantıları çalışılmıştır. Hiperharmonik fonksiyonun trigonometrik fonksiyonlarla ilişkileri elde edilmiş, sonsuz çarpım gösterimi, integral gösterimi ve bazı türevsel özdeşlikleri verilmiştir.

Article soumis au Journal of Mathematical Analysis and Apllications (Elsevier).; By employing the Ramanujan summation method, we derive some unusual formulas for a class of Euler sums (including divergent Euler sums) similar to the classical relations due to Euler.

Article soumis au Lithuanian Mathematical Journal; By applying the Ramanujan summation method to a certain class of Euler sums, we derive some unusual formulas similar to the classical relations due to Euler.

Article soumis au Journal of Mathematical Analysis and Apllications (Elsevier).; Using the Ramanujan summation method, we derive some unusual formulas for a class of Euler sums (including divergent Euler sums) similar to the classical relations due to Euler.

Article soumis au Ramanujan Journal (Springer).; In this study, we apply the Ramanujan summation method to a certain class of Euler sums and provide new reflection formulas that extend the well-known relations of symmetry between reciprocal linear Euler sums.

By using appropriate Padé approximations to the logarithm, we obtain some reccurence formulas for Bernoulli polynomials of the second kind. The proofs are based on suitable computations of some finite hypergeometric sums and the method of generating functions.

In a recent paper [Montes Taurus J. Pure Appl. Math. 3 (1) (2021), 38–61] we defined the class of combinatorial sums. The purpose of this paper is to give some integral formulas, identities and combinatorial sums using the numbers y(n, λ). The obtained results are related to the Bernoulli numbers and their interpolation functions, as well as the Pell numbers, the Harmonic numbers, the al ternating Harmonic numbers, the Daehee numbers, and the Catalan-Qi numbers. Moreover, we give answers to some open problems involving the numbers y(n, λ). Bulletin t. 154 de l'Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences mathematiques no 46.

In this study, we apply the Ramanujan summation method to linear Euler sums. This enables us to establish new reflection formulas for these sums that complete results already achieved in our previous works.; Dans cette étude, on applique le procédé de sommation de Ramanujan aux sommes d'Euler linéaires. Ceci nous permet d'établir de nouvelles relations de réciprocité entre sommes d'Euler parachevant les résultats déjà obtenus dans des travaux antérieurs.

Submitted by Boris Flegr (boris@uerj.br) on 2020-11-08T17:22:47Z No. of bitstreams: 1 Dissertacao_Thigo Cabral.pdf: 1479331 bytes, checksum: 606afa9369cc1c727b610d110998b35b (MD5) Made available in DSpace on 2020-11-08T17:22:47Z (GMT). No. of bitstreams: 1 Dissertacao_Thigo Cabral.pdf: 1479331 bytes, checksum: 606afa9369cc1c727b610d110998b35b (MD5) Previous issue date: 2020-08-27 In this dissertation we will present Stirling numbers of the first and second types, their properties and identities, some of which will be demonstrated. In addition, the relationship of these numbers to Bell numbers, Harmonic numbers an Lah numbers will be presented. We will also partially address a problem that relates a certain differential equation to Stirling numbers. Neste trabalho iremos apresentar os números de Stirling de primeiro e segundo tipos, suas propriedades e identidades, e demonstrando algumas delas. Além disso será apresentada a relação desses números com os números de Bell, números Harmônicos e números de Lah. Também será tratatado parcialmente um problema que relaciona certa equação diferencial com números de Stirling.

International audience; In this study, we determine certain constants which naturally occur in the Laurent expansion of harmonic zeta functions.

We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors., Improvement form previous version with an addendum by J. Benatar and A. Nishry

By means of the modified Abel lemma on summation by parts, we investigate infinite series involving quadratic and cubic harmonic numbers. Several infinite series identities are established for $\pi^2$ and $\zeta(3)$ as consequences.

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 other techniques (such as the Laplace and other types of integral transforms) yield a very efficient tool to explore the properties of these numbers.

International audience; In this article, we study a class of conditionally convergent alternating series including, as a special case, the famous series $\sum_{n\geq 2}(−1)^n \frac{ζ(n}{n}$ which links Euler’s constant $\gamma$ to special values of the Riemann zeta function at positive integers. We give several new relations of the same kind. Among other things, we show the existence of a similar relation for the Apostol-Vu harmonic zeta function which have never been noticed before. We also highlight a deep connection with the Ramanujan summation of certain divergent series which originally motivated this work.

The Lagrange expansion formula is employed to determine the Maclaurin series for the logarithms of Lambert series of binomial coefficients, extending the log–squared of the Catalan generating function due to Knuth

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 H. Chen (2010)., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1309.7560

In this paper, with helps of some combinatorial identities, we investigate various basic congruences involving harmonic numbers and terms of the second order sequences {Ukn} and {Vkn}.

Bu makalede harmonik sayıları ve ikinci mertebeden lineer dizilerinterimlerini içeren toplamlar hakkında bazı denklikler gösterilmiştir., In this paper, we establish some congruences involving sums withharmonic numbers and the terms of second-order linear sequences.

The principal purpose of this paper is to present and implement two numerical algorithms for solving linear and nonlinear fifth-order two point boundary value problems. These algorithms are developed via establishing a new Galerkin operational matrix of derivatives. The nonzero elements of the derived operational matrix are expressed explicitly in terms of the well-known harmonic numbers. The key idea for the two proposed numerical algorithms is based on converting the linear or nonlinear fifth-order two BVPs into systems of linear or nonlinear algebraic equations by employing Petrov-Galerkin or collocation spectral methods. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the two proposed algorithms.

Работа посвящена описанию закономерностей аппроксимации частичной суммы обобщенного гармонического числового ряда. Ряды этого типа достаточно часто используются в теории чисел, в описании физических закономерностей и в вычислительной математике. В работе предложен вариант аналитической оценки частичной суммы обобщенного гармонического ряда форме Эйлера. То есть, частичная сумма равна значению интеграла от общего члена ряда для последнего учтенного слагаемого, плюс постоянная, плюс бесконечно-малая функция., The paper is devoted to the description of approximation regularities of the partial sum of the generalized harmonic number series. Series of this type are often used in number theory, in the description of physical laws and computational mathematics. An option of the analytical estimation of the partial sum of the generalized harmonic series in the Eulerian form is proposed. That is, the partial sum is equal to the value of the general term integral of the series for the last counted term, plus a constant, plus an infinitesimal function., №04(70) (2018)

International audience; In this article, new series for the first and second Stieltjes constants (also known as generalized Eu-ler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so–called Gregory coefficients, which are also known as (reciprocal) logarithmic numbers, Cauchy numbers of the first kind and Bernoulli numbers of the second kind. In addition, two interesting series with rational terms for Euler's constant γ and the constant ln 2π are given, and yet another generalization of Euler's constant is proposed and various formulas for the calculation of these constants are obtained. Finally, we mention in the paper that almost all the constants considered in this work admit simple representations via the Ramanujan summation.

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 ? ( 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.

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. Romanian National Authority for Scientific Research/[PN-II-ID-PCE-2011–3-0087]//Rumanía Universidad de Costa Rica/[]/UCR/Costa Rica UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de investigaciones Matemáticas y Metamatemáticas (CIMM)

WOS: 000386418700026, It is an open question asked by Mezo 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. (C) 2016 Elsevier Inc. All rights reserved., Nesin Mathematics Village, The authors are very grateful to the Nesin Mathematics Village for their support and warm hospitality during this work.

We prove the following very accurate approximation formula for the factorial function: This gives better results than the following approximation formula which is established by the author [5] and C. Mortici [16] independently, and gives similar results with which is established by C. Mortici in his very new paper [8].

There is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijovic, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear.

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.

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 ⌋ .

Three new closed-form summation formulae involving harmonic numbers are established using simple arguments and they are very general extensions of Euler's famous harmonic sum identity. Some illustrative special cases as well as immediate consequences of the main results are also considered.

Recently, Srivastava et al. (2011) [2] unified and extended several interesting generalizations of the familiar Hurwitz–Lerch Zeta function Φ ( z , s , a ) by introducing a Fox–Wright type generalized hypergeometric function in the kernel. For this newly introduced special function, two integral representations of different kinds are investigated here by means of a known result involving a Fox–Wright generalized hypergeometric function kernel, which was given earlier by Srivastava et al. (2011) [2] , and by applying some Mathieu ( a , λ ) -series techniques. Finally, by appealing to each of these two integral representations, two sets of two-sided bounding inequalities are proved, thereby extending and generalizing the recent work by Jankov et al. (2011) [15] .

In this paper we prove new complete monotonicity properties of some functions involving the gamma function. As consequences of them we establish various new upper and lower bounds for the gamma function and the harmonic numbers. (C) 2011 Elsevier Ltd. All rights reserved.

Let the numbers P(r,n,k) be defined by P(r,n,k):=P"r(H"n^(^1^)-H"k^(^1^),...,H"n^(^r^)-H"k^(^r^)), where P"r(x"1,...,x"r)=(-1)^rY"r(-0!x"1,-1!x"2,...,-(r-1)!x"r) and Y"r are the exponential complete Bell polynomials. By observing that the numbers P(r,n,k) generate two Riordan arrays, we establish several general summation formulas, from which series of harmonic number identities are obtained. In particular, some of these harmonic number identities also involve other special combinatorial sequences, such as the Stirling numbers of both kinds, the Lah numbers, the Bernoulli numbers and polynomials and the Cauchy numbers of both kinds.

Let γ = 0.577215 … be the Euler–Mascheroni constant, and let R n = ∑ k = 1 n 1 k − log ( n + 1 2 ) . We prove that for all integers n ≥ 1 , 1 24 ( n + a ) 2 ≤ R n − γ 1 24 ( n + b ) 2 with the best possible constants a = 1 24 [ − γ + 1 − log ( 3 / 2 ) ] − 1 = 0.55106 … and b = 1 2 . This refines the result of D. W. DeTemple, who proved that the double inequality holds with a = 1 and b = 0 .

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli-Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form. (C) 2009 Elsevier Inc. All rights reserved.

In this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.

In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.