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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.

In this paper, we gave some new identities and super-congruences involving binomial coefficient, and some interesting congruences related to Fibonacci and Lucas sequence. [ABSTRACT FROM AUTHOR]

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.

In this paper, we give some identities involving derangement numbers. In addition, we derive nonlinear differential equation from the generating function of r-derangement numbers and obtain some identities including generalized harmonic numbers and derangement numbers by using this differential equation. For example, for any positive integers N, n and r, ... [ABSTRACT FROM AUTHOR]

Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: H n ∼ γ + log n − ∑ k = 1 ∞ B k k · n k , where B k is the Bernoulli numbers. In this paper, we rewrite Ramanujan's harmonic number expansion into a similar form of Euler's asymptotic expansion as n approaches infinity: H n ∼ γ + c 0 (h) log (q + h) − ∑ k = 1 ∞ c k (h) k · (q + h) k , where q = n (n + 1) is the nth pronic number, twice the nth triangular number, γ is the Euler–Mascheroni constant, and c k (x) = ∑ j = 0 k k j c j x k − j , with c k is the negative of the median Bernoulli numbers. Then, 2 c n = ∑ k = 0 n n k B n + k , where B n is the Bernoulli number. By using the result obtained, we present two general Ramanujan's asymptotic expansions for the nth harmonic number. For example, H n ∼ γ + 1 2 log (q + 1 3) − 1 180 (q + 1 3) 2 ∑ j = 0 ∞ b j (r) (q + 1 3) j 1 / r as n approaches infinity, where b j (r) can be determined. [ABSTRACT FROM AUTHOR]

Let s 0 , s 1 , s 2 , ... be a sequence of rational numbers whose m th divided difference δ m s is integer-valued. We prove that s n is a polynomial function in n if | s n | ≪ θ n for some positive number θ satisfying θ < e 1 + 1 2 + ⋯ + 1 m − 1. [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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the following conjectures of Z.-H. Sun [New congruences involving Apéry-like numbers, preprint (2020). Available at arXiv:2004.07172v2]: Let p>3 be a prime. Then G 2 p ≡ G 2 + 3072 (− 1) (p − 1) / 2 p 2 E p − 3 (mod p 3) , G 2 p − 1 ≡ (− 1) (p − 1) / 2 16 4 (p − 1) G 1 + 164 p 2 E p − 3 (mod p 3) , G 3 p ≡ G 3 + 94464 (− 1) p − 1 2 p 2 E p − 3 (mod p 3). where G n = ∑ k = 0 n ( 2 k k ) 2 ( 2 n − 2 k n − k ) 4 n − k , and E n stands for the nth Euler number. [ABSTRACT FROM AUTHOR]

We prove three supercongruences for sums of the Almkvist–Zudilin numbers, which confirm some conjectures of Zudilin and Sun. A typical example is the Ramanujan-type supercongruence: ∑ k = 0 p - 1 4 k + 1 81 k γ k ≡ - 3 p p (mod p 3) , which is corresponding to Chan and Verrill's formula for 1 / π : ∑ k = 0 ∞ 4 k + 1 81 k γ k = 3 3 2 π. Here γ n are the Almkvist–Zudilin numbers. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the following conjectures of Z.-H. Sun [New congruences involving Apéry-like numbers, preprint (2020). Available at arXiv:2004.07172v2]: Let p>3 be a prime. Then G 2 p ≡ G 2 + 3072 (− 1) (p − 1) / 2 p 2 E p − 3 (mod p 3) , G 2 p − 1 ≡ (− 1) (p − 1) / 2 16 4 (p − 1) G 1 + 164 p 2 E p − 3 (mod p 3) , G 3 p ≡ G 3 + 94464 (− 1) p − 1 2 p 2 E p − 3 (mod p 3). where G n = ∑ k = 0 n ( 2 k k ) 2 ( 2 n − 2 k n − k ) 4 n − k , and E n stands for the nth Euler number. [ABSTRACT FROM AUTHOR]

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 1p + 2p + + np 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.

Let L be the set of all positive integers n such that the denominator of 1+1/2+···+1/n is less than the least common multiple of 1, 2,...,n. In this paper, under a certain assumption on linear independence, we prove that the set L has the upper asymptotic density 1. The assumption follows from Schanuel’s conjecture. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers h n r 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. [ABSTRACT FROM AUTHOR]

In this paper, we present the notion of a magic pair of digits in a number system with an arbitrary basis b, and we study its equivalent definitions. In particular, we determine all pairs of magic digits in decimal, octal, hexadecimal, and binary number system. We use magic pairs of digits to derive a new multiplication formula. In the end, we give an application of a well-known formula related to magic pairs. [ABSTRACT FROM AUTHOR]

We show that the formula of Faà di Bruno for the nth derivative of a composite function can be applied to deduce various combinatorial identities involving Lah numbers, Stirling numbers and harmonic numbers. One of our results states that for all integers m, n with m ≥ n ≥ 1 , we have 2 n m n (2 H 2 m - n - H m - n) = ∑ k = 0 n / 2 (- 1) k m k 2 m - 2 k n - 2 k (2 H 2 (m - k) - H m - k) , where H k = 1 + 1 / 2 + ⋯ + 1 / k denotes the kth harmonic number. Moreover, we present connections between absolutely and completely monotonic functions and we provide a three-parameter class of completely monotonic functions. [ABSTRACT FROM AUTHOR]

Recently the first author proved a congruence proposed in 2006 by Adamchuk: ∑ k = 1 ⌊ 2 p / 3 ⌋ 2 k k ≡ 0 (mod p 2) for any prime p = 1 (mod 3). In this paper, we provide more examples (with proofs) of congruences of the same kind ∑ k = 1 ⌊ a p / r ⌋ 2 k k x k (mod p 2) where p is a prime such that p ≡ 1 (mod r) , a / r is a fraction in (1 / 2 , 1) and x is a p -adic integer. The key ingredients are the p -adic Gamma function Γ p and a special class of computer-discovered hypergeometric identities. [ABSTRACT FROM AUTHOR]

For any number field, we define Dedekind harmonic numbers with respect to this number field. First, we show that they are not integers except finitely many of them. Then, we present a uniform and an explicit version of this result for quadratic number fields. Moreover, by assuming the Riemann hypothesis for Dedekind zeta functions, we prove that the difference of two Dedekind harmonic numbers are not integers after a while if we have enough terms, and we prove the non-integrality of Dedekind harmonic numbers for quadratic number fields in another uniform way together with an asymptotic result. [ABSTRACT FROM AUTHOR]

The values of four improper integrals containing the squares of the tails of the Maclaurin series for the sine and cosine functions are computed using standard residue theory for contour integrals. Using a very different approach, we then provide solutions to some open questions concerning two related improper integrals. [ABSTRACT FROM AUTHOR]

In this paper, we give a generalization of Euler's pentagonal number theorem. As an application, we derive a recursion formula for successively determining the coefficients ak of 1=nk. We conclude this paper by presenting brief descriptions for some related results and other motivating developments. [ABSTRACT FROM AUTHOR]

Let p be a prime greater than or equal to 5: In this paper, by using the harmonic numbers and Fermat quotient we establish congruences involving the sums... [ABSTRACT FROM AUTHOR]

In this paper, we establish some congruences involving the trinomial coefficients and harmonic numbers. For example, for any prime p > 3, ∑p−1 k=0 (−1)k ( p − 1 k ) 2 Hk ≡ 0 (mod p). [ABSTRACT FROM AUTHOR]

Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers. In this paper, we rewrite Ramanujan’s harmonic number expansion into a similar form of Euler’s asymptotic expansion as n approaches infinity: Hn∼γ+c0(h)log(q+h)−∑k=1∞ck(h)k·(q+h)k, where q=n(n+1) is the nth pronic number, twice the nth triangular number, γ is the Euler–Mascheroni constant, and ck(x)=∑j=0kkjcjxk−j, with ck is the negative of the median Bernoulli numbers. Then, 2cn=∑k=0nnkBn+k, where Bn is the Bernoulli number. By using the result obtained, we present two general Ramanujan’s asymptotic expansions for the nth harmonic number. For example, Hn∼γ+12log(q+13)−1180(q+13)2∑j=0∞bj(r)(q+13)j1/r as n approaches infinity, where bj(r) can be determined.

This paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the Riemann hypothesis. We give an account on the theorem of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the series 𝒨is(z)=∑n=1∞μ(n)nszn$\mathcal{M}{i_s}(z) = \sum\limits_{n = 1}^\infty {{{\mu (n)} \over {{n^s}}}{z^n}} $. A significant result is that the real part f of

By means of the generating function approach, we derive several summation formulae involving multiple harmonic numbers H ,,,/■(a), as well as other combinatorial numbers named after Bernoulli, Euler, Bell, Genocchi and Stirling. [ABSTRACT FROM AUTHOR]

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function η(s), and hence Riemann's function ζ(s), is obtained in terms of the Exponential Integral function Es(iκ) of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions ζ±(s) and η±(s) are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints. [ABSTRACT FROM AUTHOR]

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.

We evaluate Euler-like sums involving harmonic numbers using expansions for powers of logarithmic, arctan and a r c t a n h functions. [ABSTRACT FROM AUTHOR]

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Flajolet and Salvy pointed out that every Euler sum is a Q -linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and compositions, we establish two explicit formulas for the Euler sums, and show that all the Euler sums are indeed expressible in terms of MZVs. Moreover, we apply this method to the alternating Euler sums, and show that all the alternating Euler sums are reducible to alternating MZVs. Some famous results, such as the Euler theorem, the Borwein–Borwein–Girgensohn theorems, and the Flajolet–Salvy theorems can be obtained directly from our theory. Some other special cases, such as the explicit expressions of S r m , q , S r m , q ¯ , S r ¯ m , q and S r ¯ m , q ¯ , are also presented here. The corresponding Maple programs are developed to help us compute all the sums of weight w ≤ 11 for non-alternating case and of weight w ≤ 6 for alternating case. [ABSTRACT FROM AUTHOR]

By using the Wilf–Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for ζ (2 + r + 2 s). Such identity, which is an extension of a Bailey–Borwein–Bradley Apéry-like formula for even zeta values, is then applied to show several supercongruences. [ABSTRACT FROM AUTHOR]

In this study, depending on the upper and the lower indices of the hyperharmonic number h(r)n, 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. [ABSTRACT FROM AUTHOR]

We give a rational approximation to Euler's constant at a geometric rate of convergence, which is easy to compute. Moreover, such an approximation is completely monotonic. The approximants are built up in terms of expectations of the harmonic numbers acting on the standard Poisson process. [ABSTRACT FROM AUTHOR]

In this paper, we study the explicit formulas of some Euler sums and Stirling sums, which are infinite series on harmonic numbers and Stirling numbers, respectively. As a result, we show that some Stirling sums are expressible in terms of special integrals and alternating multiple zeta values, and all the Euler sums involving negative powers of two are expressible in terms of multiple polylogarithms, hence in terms of unit-exponent alternating multiple zeta values. Some special cases are discussed, and some identities on Euler sums and alternating MZVs are obtained, including a conjectural one due to Borwein et al. Moreover, the Maple program based on the explicit formula is developed, so that the Euler sums involving negative powers of two and of weight w ≤ 7 can be computed automatically. [ABSTRACT FROM AUTHOR]

In a recent article, the authors have studied the arithmetic natures of a certain class of harmonic numbers. The present paper will explore the linear independence of harmonic numbers and establish the results to a more general case. More precisely, we obtain the dimension of space generated by these harmonic numbers for primes and their powers. In fact, for the case when an integer is divisible by only two distinct primes say m = p 1 α 1 p 2 α 2 , define W m = Q ¯ - span of { H a / m | 1 ≤ a ≤ m }. We prove that if m satisfies some necessary conditions, then dim Q ¯ W m ≤ ϕ (m) 2 + 4. Moreover, we generalize the result to any finite set of integers of the form m = p 1 α 1 p 2 α 2 under the assumption of the same necessary conditions. [ABSTRACT FROM AUTHOR]

The Gauss summation theorem and an extended 3 F 2 -series of Watson and Whipple type are examined by means of power series expansions. Numerous Ramanujan-like series involving harmonic numbers and squared central binomial coefficients are evaluated in closed forms. Several remarkable identities discovered by Campbell (Ramanujan J 46(2):373–387, 2018) are included as particular cases. [ABSTRACT FROM AUTHOR]

We study the properties of a sequence c n defined by the recursive relation c 0 n + 1 + c 1 n + 2 + ⋯ + c n 2 n + 1 = 0 for n ≥ 1 and c 0 = 1. This sequence also has an alternative definition in terms of certain norm minimization in the space L 2 ([ 0 , 1 ]). We prove estimates on the growth order of c n and the sequence of its partial sums, infinite series identities, connecting c n with the harmonic numbers H n and also formulate some conjectures based on numerical computations. [ABSTRACT FROM AUTHOR]

We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents of these continued fraction expansions. Linear fractional transformations of such continued fractions are also discussed. We show more continued fraction expansions for different numbers and types, in particular, on Cauchy numbers. [ABSTRACT FROM AUTHOR]

Let Hn be the n-th harmonic number and let vn be its denominator. It is known that vn is even for every integer n > = 2. In this paper, we study the properties of Hn and prove that for any integer n, vn = en(1+o(1)). In addition, we obtain some results of the logarithmic density of harmonic numbers. [ABSTRACT FROM AUTHOR]

Congruences involving sums of Harmonic numbers and binomial coefficients are considered in this paper. Recently, many great mathematicians have been interested to find congruences and relationships between these numbers such Sun & Tauraso, Koparal & Ömür, Mao & Sun and Meštrović & Andjić. In the present paper, some new combinatorial congruences are proved. These congruences are mainly determined modulo p² or p³ (in any prime) and they are motivated by a recent paper by Meštrović and Andjić. The first main result (Theorem 1) presents the congruence modulo p² (p > 3 is any prime) involving sum of products of two binomial coefficients and Harmonic numbers. Two interesting congruences modulo a prime p > 3 (Corollary 2) involving Harmonic numbers Hk, Catalan numbers ck and Fermat quotient q2 := (2p-1 - 1)/p are obtained as consequences of Theorem 1. The second main result (Theorem 2) presents the congruence modulo p³(p > 3 is any prime) involving sum of products of two binomial coefficients and Harmonic numbers. [ABSTRACT FROM AUTHOR]

Let {L(t), t ≥ 0} be a Lévy process with representative random variable L(1) defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Lévy measure of this process. We also define two subordinated processes. The first one, Y(t), is a Negative-Binomial process X(t) directed by Gamma process. The second process, Z(t), is a Logarithmic Lévy process L(t) directed by Poisson process. For them, we prove that the Bernstein functions of the processes L(t) and Y(t) contain the iterated logarithmic function. In addition, the Lévy measure of the subordinated process Z(t) is a shifted Lévy measure of the Negative-Binomial process X(t).We compare the properties of these processes, knowing that the total masses of corresponding Lévy measures are equal. [ABSTRACT FROM AUTHOR]

In this paper, we prove three congruences involving Apéry-like numbers D n , S n and T n which have been conjectured by Sun. These numbers are defined by D n = ∑ k = 0 n n k 2 2 k k 2 n − 2 k n − k , S n = ∑ k = 0 n n k 2 k k 2 n − 2 k n − k , T n = ∑ k = 0 n n k 2 2 k n 2. [ABSTRACT FROM AUTHOR]