Search

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.

We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa–Kaneko zeta function, studied before amongst others by Candelpergher and Coppo, and also by Young. We complement and generalize earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values. [ABSTRACT FROM AUTHOR]

Abstract Using the software package Sigma developed by Schneider, we automatically discover and prove some combinatorial identities involving harmonic numbers, from which we deduce some supercongruences on partial sums of hypergeometric series. These results confirm some conjectural generalizations of van Hamme's supercongruences in some special cases, which were recently proposed by Guo (2017). [ABSTRACT FROM AUTHOR]

Abstract Text Let v n be the denominator of the n -th harmonic number H n = 1 + 2 − 1 + ⋯ + n − 1. In this paper, we prove the following unusual result: the set of positive integers n with v n = v n + 1 has density one. We also prove that the same result is true for the denominators of alternating harmonic numbers. Video For a video summary of this paper, please visit https://youtu.be/Sk0zMSKPe3M. [ABSTRACT FROM AUTHOR]

In this paper, by the Faà di Bruno formula, we establish the decompositions of two general fractions involving the reciprocals of products of binomial coefficients. Using the decompositions, we discuss the evaluations of some Euler-type sums involving harmonic numbers and binomial coefficients, such as and some other forms. We present some explicit evaluations as examples and provide the Maple package to compute the sums and. It can be found that this work gives a unified approach to such sums and generalizes many known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we extend tools developed in [26] to study harmonic numbers sums. We use these objects to compute Euler-related sums and to explore their connections with classical Euler sums. We establish some new relations between Euler sums and related sums. The relations obtained allow us to find some nice closed-form representations of nonlinear Euler sums through zeta values and linear sums. [ABSTRACT FROM AUTHOR]

By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the form ... for p = 1 and 2, satisfy certain recurrence relations which allow us to write them in terms of zeta values, polylogarithms and ln 2. Furthermore, we also obtain reductions for certain multiple polylogarithmic values at 1/2. [ABSTRACT FROM AUTHOR]

We develop new closed form representations of sums of reciprocal binomial coefficients. We also identify new integral and hypergeometric representation for the binomial-harmonic number sums. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the integral of x n log p (sin (x)) for natural numbers n and p. In doing so, we recover some well-known results and remark on some relations to the log-sine integral Ls n + p + 1 (n) (θ). Later, we use properties of Bell polynomials to find an expression for the derivative of the central binomial and shifted central binomial coefficients as finite sums of polygamma functions and harmonic numbers. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we investigate some properties of the Hyperharmonic function defined by ... Using this definition we introduce harmonic numbers with complex index and we give some series of these numbers. Also formulas for the calculation of harmonic numbers with rational index are obtained. For the simplicity of differentiation we reorganized representation of H(w) z. With the help of this new form we get higher order derivatives of the Hyperharmonic function more easily. Besides these, owing to the fact that the Hyperharmonic function is composed of some important functions, we interested in properties and connections of it. We get connections between the Hyperharmonic function and trigonometric functions. Infinite product representation, integral representation and differentiation identities of this function are also obtained. [ABSTRACT FROM AUTHOR]

We present a number of identities in the form of Stirling transforms. These identities involve Stirling numbers of the first and second kinds, hyperharmonic and derangement numbers, Bernoulli and Euler numbers and polynomials, powers, power sums, and factorials. [ABSTRACT FROM AUTHOR]

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 (2015). [ABSTRACT FROM AUTHOR]

In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers in terms of harmonic numbers. As consequences of this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind. We also give a relationship between the p-Bernoulli numbers and the generalized Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

Abstract 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, 2010).

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}.

By applying the derivative operator to Whipple-type 3 F 2 -series identities and utilizing L'Hôspital rule, two families of summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM AUTHOR]

In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta function. Moreover, we provide explicit evaluations for almost all Euler sums with weight ≤5, which can be expressed in terms of zeta values and polylogarithms. Furthermore, we give explicit formula for several classes of Euler-related sums in terms of zeta values and harmonic numbers, and several examples are given. The given representations are new. [ABSTRACT FROM AUTHOR]

Let a , n , m ∈ ℤ + with (a , p) = 1 and p be an odd prime. We find a supercongruence for ∑ p | k a p n k m and related sums of powers of binomial coefficients. These results complement prior results for ∑ k ≡ i (mod   p) a p n k m with i = 1 , 2 , … , p − 1 obtained recently by the author. [ABSTRACT FROM AUTHOR]

By applying the derivative operator to Watson-type -series identities and using the bijection method of two-term difference as well as L'Hôspital rule, three families of summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM AUTHOR]

We provide various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated. [ABSTRACT FROM AUTHOR]

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 effciency and accuracy of the two proposed algorithms. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to present and analyze a numerical algorithm for the solution of eighth-order boundary value problems. The proposed solutions are spectral and they depend on a new operational matrix of derivatives of certain shifted Legendre polynomial basis, along with the application of the collocation method. The nonzero elements of the operational matrix are expressed in terms of the well-known harmonic numbers. Numerical examples provide favorable comparisons with other existing methods and ascertain the efficiency and applicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime $$p>3$$ , we determine modulo $$p^2$$ , where $$C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right) $$ is the k-th Catalan number and $$C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right) $$ is the second-order Catalan numbers of the first kind. And we prove that where $$D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) $$ is the n-th Delannoy number and $$q_p(2)=(2^{{p-1}}-1)/p$$ is the Fermat quotient. [ABSTRACT FROM AUTHOR]

Consider an ordinary generating function ∑ k = 0 ∞ c k x k , of an integer sequence of some combinatorial relevance, and assume that it admits a closed form C ( x ) . Various instances are known where the corresponding truncated sum ∑ k = 0 q − 1 c k x k , with q a power of a prime p , also admits a closed form representation when viewed modulo p . Such a representation for the truncated sum modulo p frequently bears a resemblance with the shape of C ( x ) , despite being typically proved through independent arguments. One of the simplest examples is the congruence ∑ k = 0 q − 1 ( 2 k k ) x k ≡ ( 1 − 4 x ) ( q − 1 ) / 2 ( mod p ) being a finite match for the well-known generating function ∑ k = 0 ∞ ( 2 k k ) x k = 1 / 1 − 4 x . We develop a method which allows one to directly infer the closed-form representation of the truncated sum from the closed form of the series for a significant class of series involving central binomial coefficients. In particular, we collect various known such series whose closed-form representation involves polylogarithms Li d ( x ) = ∑ k = 1 ∞ x k / k d , and after supplementing them with some new ones we obtain closed-forms modulo p for the corresponding truncated sums, in terms of finite polylogarithms £ d ( x ) = ∑ k = 1 p − 1 x k / k d . [ABSTRACT FROM AUTHOR]

In terms of Abel's transformation on difference operators, we establish four families of summation formulas involving generalized harmonic numbers. They include several known and numerous new harmonic number identities as special cases. [ABSTRACT FROM AUTHOR]

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.

Let be an even integer and be an odd prime. We study Franel-like sums and alternating sums, as well as lacunary sums of th powers of binomial coefficients from the point of view of arithmetic properties. This paper complements the author's prior work on the cases with odd although, it uses a different approach. It develops new supercongruences and determines the -adic order of these sums as well as of generalized harmonic sums restricted to particular remainder classes modulo . [ABSTRACT FROM AUTHOR]

In this paper we mainly prove the following two conjectures of Z.-W. Sun [10] : For any odd prime p , we have ∑ k = 0 p − 1 P k 8 k ≡ 1 + 2 ( − 1 ) ( p − 1 ) / 2 p 2 E p − 3 ( mod p 3 ) , ∑ k = 0 p − 1 P k 16 k ≡ ( − 1 ) ( p − 1 ) / 2 − p 2 E p − 3 ( mod p 3 ) , where P n = ∑ k = 0 n ( 2 k k ) 2 ( 2 ( n − k ) n − k ) 2 ( n k ) is the n -th Catalan–Larcombe–French number, E n are the Euler numbers which are defined by E 0 = 1 , E n = − ∑ k = 1 ⌊ n / 2 ⌋ ( n 2 k ) E n − 2 k ( n ≥ 1 ) . [ABSTRACT FROM AUTHOR]

In this paper, we show that for any prime , and where denotes the Bernoulli polynomial of degree . And we prove that and where stands for the th Bernoulli number. This confirms several conjectures of Z.-W. Sun. [ABSTRACT FROM AUTHOR]

For any we first give new proofs for the following well-known combinatorial identities and and then we produce the generating function and an integral representation for . Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that and where are generalized harmonic numbers defined below. [ABSTRACT FROM AUTHOR]

Copyright of Karaelmas Science & Engineering Journal / Karaelmas Fen ve Mühendislik Dergisi is the property of Karaelmas Science & Engineering Journal and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Text Let H n be the n -th harmonic number and let u n be its numerator. For any prime p , let J p be the set of positive integers n with p | u n . In 1991, Eswarathasan and Levine conjectured that J p is finite for any prime p . It is clear that the p -adic valuation of H n is not less than − ⌊ log p ⁡ n ⌋ . Let T p be the set of positive integers n such that the p -adic valuation of H n is equal to − ⌊ log p ⁡ n ⌋ . Recently, Carlo Sanna proved that | J p ∩ [ 1 , x ] | < 129 p 2 / 3 x 0.765 and that there exists S p ⊆ T p with δ ( S p ) > 0.273 , where δ ( X ) denotes the logarithmic density of the set X of positive integers. He also commented that with his methods δ ( S p ) > 1 / 3 − ε cannot be achieved. In this paper, we improve these results. For example, two of our results are: (a) | J p ∩ [ 1 , x ] | ≤ 3 x 2 / 3 + 1 / ( 25 log ⁡ p ) ; (b) δ ( T p ) exists and 1 − ( 2 log ⁡ p ) − 1 ≤ δ ( T p ) ≤ 1 − ( p log ⁡ p ) − 1 for all primes p ≥ 13 . In particular, δ ( T p ) > 0.63 for all primes p . Video For a video summary of this paper, please visit https://youtu.be/3ujCuVwH8k8 . [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, by the Riordan array method, we show that in Ramanujan's asymptotic expansion of the harmonic numbers, the coefficients given by Villarino satisfy the recurrence given by Chen and Cheng. This gives an affirmative answer to Chen and Cheng's open problem. [ABSTRACT FROM AUTHOR]

In terms of the derivative operator, integral operator and a Saalschütz-type identity, two families of summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some formulae related to Euler Sums and give the closed-form representations for the sumwhich can be evaluated in terms of the Riemann zeta functions and generalized harmonic numbers. [ABSTRACT FROM AUTHOR]

In this paper, we develop an approach to evaluation of nonlinear Euler sums. The approach is based on Tornheim type series computations. By the approach, we can obtain some closed form representations of quadratic and cubic sums in terms of zeta values and linear sums. Furthermore, we also evaluate several other series involving harmonic numbers. Some interesting new consequences and illustrative examples are considered. [ABSTRACT FROM AUTHOR]