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Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions. [ABSTRACT FROM AUTHOR]

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz's zeta function derivative. [ABSTRACT FROM AUTHOR]

By employing the coefficient extraction method, a class of binomial series involving harmonic numbers will be reviewed through three hypergeometric F 1 2 (y 2) -series. Numerous closed-form generating functions for infinite series containing binomial coefficients and harmonic numbers will be established, including several conjectured ones. [ABSTRACT FROM AUTHOR]

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 for charging each battery is known, whereas it is assumed that the power drops to zero at the moment when the battery gets fully loaded. The power usage function is linear and decreasing. The charging jobs are nonpreemptable and independent, whereas the total available amount of power is limited. The objective is to minimize the schedule length. In this paper, we analyze the case of a problem with identical jobs that already cover a wide variety of practical situations. By employing inverses of natural numbers, similar to harmonic series, we prove two properties of this case, and we also discuss the phenomenon of the stabilization of the difference between the start times of two successive jobs in a schedule. We also take under examination a few special cases of the problem. Some conclusions and directions for future research are given. [ABSTRACT FROM AUTHOR]

We determine the values of several infinite series involving the Riemann zeta function. In particular, degree one rationally weighted summands involving the Riemann function evaluated at even numbers give a finite sum involving only the Riemann function evaluated at odd numbers. [ABSTRACT FROM AUTHOR]

Yamamoto's integral was the integral associated with 2-posets, which was first introduced by Yamamoto. In this paper, we obtained the values of infinite series involving harmonic numbers and reciprocal of binomial coefficients by using some techniques of Yamamoto's integral. We determine the value of infinite series of the form: $ \sum\limits_{m_1,\ldots,m_n,\ell_1,\ldots,\ell_k\geq 1}\frac{H_{m_1}^{(a_1)}\cdots H_{m_n}^{(a_n)}} {m_1^{b_1}\cdots m_n^{b_n}\ell_1^{c_1}\cdots\ell_k^{c_k} \binom{m_1+\cdots+m_n+\ell_1+\cdots+\ell_k}{\ell_k}}, $ in terms of a finite sum of multiple zeta values, for positive integers $ a_1, \ldots, a_n, b_1, \ldots, b_n, c_1, \ldots, c_k $.

In this paper, we solve the open problem posed by Kuba by expressing ∑ j , k ≥ 1 H k (u) H j (v) H j + k (w) j r k s (j + k) t as a linear combination of multiple zeta values. These sums include Tornheim's double series as a special case. Our approach is based on employing two distinct methods to evaluate the specific integral proposed by Yamamoto, which is associated with the two-poset Hasse diagram. We also provide a new evaluation formula for the general Mordell–Tornheim series and some similar types of double and triple series. [ABSTRACT FROM AUTHOR]

In this paper, we define new sequence Sn(a,b) with parameters a and b with the help of the generalized harmonic numbers. Also, we get some new sums involving harmonic, Fibonacci and Lucas numbers. [ABSTRACT FROM AUTHOR]

By applying the derivative operator to the known identities from hypergeometric series or WZ pairs, we obtain six series associated with harmonic numbers. Specifically, five of them are Ramanujan-like formulas for $ 1/\pi $ 1 / π and the remaining one contains harmonic numbers of order 2. As conclusions, Sun's four conjectural series are proved. [ABSTRACT FROM AUTHOR]

We explore a conjecture posed by Eswarathasan and Levine on the distribution of p-adic valuations of harmonic numbers H (n) = 1 + 1 / 2 + ⋯ + 1 / n that states that the set J p of the positive integers n such that p divides the numerator of H(n) is finite. We proved two results, using a modular-arithmetic approach, one for non-Wolstenholme primes and the other for Wolstenholme primes, on an anomalous asymptotic behaviour of the p-adic valuation of H (p m n) when the p-adic valuation of H(n) equals exactly 3. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then \begin{align*} &A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\ &A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\ &A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4), \end{align*} where $A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the n th Apéry number, and Bn is the n th Bernoulli number. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove some congruences involving binomial coefficients and binary quadratic forms. One such example is the following: Let p b be a prime such that p = x 2 + 2 y 2 ≡ 1 (mod 8) . Then, p ∑ k = 0 p - 1 2 k k 2 (8 k + 1) 16 k ≡ 3 p ∑ k = 0 p - 1 2 k k 2 (8 k + 3) 16 k ≡ 4 x 2 - 2 p - p 2 4 x 2 (mod p 3). [ABSTRACT FROM AUTHOR]

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions.

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz’s zeta function derivative.

By employing the coefficient extraction method, a class of binomial series involving harmonic numbers will be reviewed through three hypergeometric F12(y2)-series. Numerous closed-form generating functions for infinite series containing binomial coefficients and harmonic numbers will be established, including several conjectured ones.

The aim of this work is to present an interesting identity relating generalized hypergeometric functions. It allows us to prove many generalizations of recent results, and also to evaluate the sum of several families of Ramanujan-type series for 1 / π , 2 / π and 3 / π . Also, several families of series involving the central binomial coefficients and Harmonic numbers are evaluated. In particular, an explicit evaluation of the series ∑ n = 0 ∞ 4 - 3 n 4 n 2 n C n + m where C n is the nth Catalan number is presented, and a conjectured expression of the sum of this series given in a recent work by Seán M. Stewart is proved to be true. [ABSTRACT FROM AUTHOR]

In this paper, we begin by introducing integral formulas related to the psi functions. Following that, we delve into specific series that involve polygamma functions, leveraging Eulerian numbers to express them as finite series of double integrals. Subsequently, we employ the findings from the second part to study variant Euler harmonic sums. Finally, we offer closedform evaluations for a number of distinct instances of these variant Euler harmonic sums. [ABSTRACT FROM AUTHOR]

In this paper, we solve the open problem posed by Kuba by expressing ∑j,k≥1Hk(u)Hj(v)Hj+k(w)jrks(j+k)t as a linear combination of multiple zeta values. These sums include Tornheim’s double series as a special case. Our approach is based on employing two distinct methods to evaluate the specific integral proposed by Yamamoto, which is associated with the two-poset Hasse diagram. We also provide a new evaluation formula for the general Mordell–Tornheim series and some similar types of double and triple series.

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers Hn(p,q) ... in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of the Riemann zeta values. [ABSTRACT FROM AUTHOR]

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 present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis's infinite product formula for π , Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums. [ABSTRACT FROM AUTHOR]

In this article we prove that all squares with sides of lengths 1 m + 1 , 1 m + 2 , ⋯ can be packed in a rectangle with a one side fixed and a relatively small area. We give a similar result for all of three Moser's Packing Problems. [ABSTRACT FROM AUTHOR]

This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.

In this paper, we evaluate in closed form several different series involving the harmonic numbers and skew-harmonic numbers. We consider two classes of series involving these sequences. One class of series involves the product of the nth harmonic or skew-harmonic number and a tail. We provide the solution to two open problems concerning these harmonic series with tails from Alina SÎntămărian and Ovidiu Furdui's book Sharpening Mathematical Analysis Skills. The other class of series is the Hardy series, which involves a logarithm and the Euler-Mascheroni constant being subtracted from the nth harmonic number. [ABSTRACT FROM AUTHOR]

In this paper, we consider meromorphic extension of the function ζ h r s = ∑ k = 1 ∞ h k r k s , Re s > r (which we call hyperharmonic zeta function) where h n (r) are the hyperharmonic numbers. We establish certain constants, denoted γ h r m , which naturally occur in the Laurent expansion of ζ h r s . Moreover, we show that the constants γ h r m and integrals involving the generalized exponential integral can be written as a finite combination of some special constants. [ABSTRACT FROM AUTHOR]

We derive explicit closed-form formulas for the standard Clausen functions C l 2 m (x) and C l 2 m − 1 (x) in terms of the Hurwitz zeta function, where m is a positive integer. [ABSTRACT FROM AUTHOR]

We evaluate in closed form a large class of power series where the coefficients are products of special numbers, or products of special numbers and Laguerre polynomials. A number of examples is presented involving harmonic, hyperharmonic, Fibonacci, Lucas, Bernoulli, Euler, and derangement numbers. We also obtain a new convolution formula for the Laguerre polynomials. [ABSTRACT FROM AUTHOR]

For real numbers p , q > 1 we consider the following family of integrals: ∫ 0 1 (x q - 2 + 1) log x mq + 1 x q + 1 d x and ∫ 0 1 (x p t - 2 + 1) log x t + 1 x pt + 1 d x. We evaluate these integrals for all m ∈ N , q = 2 , 3 , 4 and p = 2 , 3 explicitly. They recover some previously known integrals. We also compute many integrals over the infinite interval [ 0 , ∞) . Applying these results, we offer many new Euler–BBP- type sums. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the following conjectures of Sun Z-H (New congruences involving Apéry-like numbers. Preprint at arXiv:2004.07172v2): Let p > 3 be a prime. Then A 2 p - 1 ′ ≡ A 1 ′ + 200 3 p 3 B p - 3 mod p 4 , D 2 p - 1 ≡ 64 2 (p - 1) D 1 - 44 3 p 3 B p - 3 mod p 4 , where A n ′ = ∑ k = 0 n n k 2 n + k k , D n = ∑ k = 0 n n k 2 2 k k 2 n - 2 k n - k and B n stands for the nth Bernoulli number. We also give a generalization of a conjecture of Sun involving Franel numbers f n = ∑ k = 0 n n k 3 . [ABSTRACT FROM AUTHOR]

We use properties of Gegenbauer polynomials to prove that the formula ∑ k = 0 n λ k λ n - k (x + y) 2 k (x - y) 2 n - 2 k = 2 λ n ∑ k = 0 n n k n - λ - 1 / 2 k k - λ - 1 / 2 k x 2 k y 2 n - 2 k (0.1) is valid for all integers n ≥ 0 and complex numbers λ , x , y . The special case λ = - 1 / 2 was proved in 2014 by Gavrea and Ivan. We apply (0.1) and related results to deduce further identities. Among others, we present combinatorial identities involving harmonic numbers and trigonometric functions, and we prove an inequality for a certain cosine sum. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove some congruences involving Apéry-like numbers S n = ∑ k = 0 n ( n k ) ( 2 k k ) ( 2 n − 2 k n − k ) : Let p>3 be a prime. Then S p ≡ S 1 + (− 1) (p − 1) / 2 8 p 2 E p − 3 (mod p 3) , S 2 p ≡ S 2 + (− 1) (p − 1) / 2 128 p 2 E p − 3 (mod p 3) , S 2 p − 1 ≡ (− 1) (p − 1) / 2 32 2 (p − 1) S 1 + 20 p 2 E p − 3 (mod p 3) , S 3 p ≡ S 3 + (− 1) (p − 1) / 2 1440 p 2 E p − 3 (mod p 3). where E n stands for the nth Euler number. [ABSTRACT FROM AUTHOR]

We establish some identities of Euler related sums. By using these identities, we discuss the closed form representations of sums of harmonic numbers and reciprocal parametric binomial coefficients through parametric harmonic numbers, shifted harmonic numbers and Riemann zeta function with positive integer arguments. In particular, we investigate products of quadratic and cubic harmonic numbers and reciprocal parametric binomial coefficients. Some illustrative special cases as well as immediate consequences of the main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove some congruences involving Apéry-like numbers S n = ∑ k = 0 n ( n k ) ( 2 k k ) ( 2 n − 2 k n − k ) : Let p>3 be a prime. Then S p ≡ S 1 + (− 1) (p − 1) / 2 8 p 2 E p − 3 (mod p 3) , S 2 p ≡ S 2 + (− 1) (p − 1) / 2 128 p 2 E p − 3 (mod p 3) , S 2 p − 1 ≡ (− 1) (p − 1) / 2 32 2 (p − 1) S 1 + 20 p 2 E p − 3 (mod p 3) , S 3 p ≡ S 3 + (− 1) (p − 1) / 2 1440 p 2 E p − 3 (mod p 3). where E n stands for the nth Euler number. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article we study some configurations of chains of mutually tangent circles that can be inscribed inside the arbelos and we find some simple algebraical conditions relating them to certain integer sequences. [ABSTRACT FROM AUTHOR]

The factorial moments of any Markov branching process describe the behaviour of its probability generating function $F(t,s)$ in the neighbourhood of the point $s=1$. They are applied to solve the forward Kolmogorov equation for the critical Markov branching process with geometric reproduction of particles. The solution includes quickly convergent recurrent iterations of polynomials. The obtained results on factorial moments enable computation of statistical measures as shape and skewness. They are also applicable to the comparison between critical geometric branching and linear birth-death processes.

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 present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis’s infinite product formula for π, Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums.

Let $\mathcal{L}$ be the set of all positive integers $n$ such that the denominator of $1+1/2+\cdots +1/n$ is less than the least common multiple of $1, 2, \dots , n$. In this paper, under a certain assumption on linear independence, we prove that the set $\mathcal{L}$ has the upper asymptotic density $1$. The assumption follows from Schanuel’s conjecture.

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.