Related papers: New Congruences on Multiple Harmonic Sums and Bern…

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and…

In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congruences for these sums modulo a prime $p.$…

The denominators $d_n$ of the harmonic number $1+\frac12+\frac13+\cdots+\frac1n$ do not increase monotonically with~$n$. It is conjectured that $d_n=D_n={\rm LCM}(1,2,\ldots,n)$ infinitely often. For an odd prime $p$, the set…

In this paper, we confirm some congruences conjectured by V.J.W. Guo and M.J. Schlosser recently. For example, we show that for primes $p>3$, $$…

The congruences for Jacobi sums of some lower orders has been treated by many authors in the literature. In this paper we establish the congruences for Jacobi sums of order 2l^2 with odd prime l. These congruences are useful to obtain…

Let the numbers $\alpha_n,\beta_n$ and $\gamma_n$ denote \begin{align*} \alpha_n=\sum_{k=0}^{n-1}{2k\choose k},\quad \beta_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{1}{k+1}\quad\text{and}\quad \gamma_n=\sum_{k=0}^{n-1}{2k\choose…

We present a novel way of realizing the Bernoulli shift on $p$ symbols on the $p$-adic integers, where $p$ is a prime. By showing that suitably small perturbations of the shift are still Bernoulli we find many "nice" maps, such as…

In this paper, we present several explicit formulas of the sums and hyper-sums of the powers of the first (n+1)-terms of a general arithmetic sequence in terms of Stirling numbers and generalized Bernoulli polynomials.

In this note, we shall provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas.

New Mersenne conjectures. The problems of simplicity, common prime divisors and free from squares of numbers $L(n) = 2^{2n}\pm2^n\pm1$ are investigated. Wonderful formulas $gcd $ for numbers $L (n) $ and numbers repunit are proved.

The Delannoy polynomial $D_n(x)$ is defined by $$ D_n(x)=\sum_{k=0}^{n}{n\choose k}{n+k\choose k}x^k. $$ We prove that, if $x$ is an integer and $p$ is a prime not dividing $x(x+1)$, then \begin{align*} \sum_{k=0}^{p-1}(2k+1)D_k(x)^3…

Let $\{A'_n\}$ be the Ap\'ery numbers given by $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k.$ For any prime $p\equiv 3\pmod 4$ we show that $A'_{\frac{p-1}2}\equiv \frac{p^2}3\binom{\frac{p-3}2}{\frac{p-3}4}^{-2}\pmod {p^3}$. Let $\{t_n\}$ be…

A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $\mu$ is identifying all $t\in\R$ such that both \[\left\{e^{-2\pi i\lambda x}: \lambda\in\Lambda\right\}\quad\text{and}\quad \left\{e^{-2\pi…

The harmonic numbers are those $H_n=\sum_{0<k\le n}\frac1k\ (n=0,1,2,\ldots)$. In this paper we confirm over ten conjectural series identities with summands involving the binomial coefficient $\binom{4k}k$ and harmonic numbers. For example,…

In the paper, we first survey some results on inequalities for bounding harmonic numbers or Euler-Mascheroni constant, and then we establish a new sharp double inequality for bounding harmonic numbers as follows: For $n\in\mathbb{N}$, the…

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.

Using an identity arising in the known Banach probability problem on matchboxes, we prove an unexpected congruence for odd prime $p:$ for $1\leq k\leq \frac{p-1}{2},\enskip \sum_{i=1}^{p-2k-1}2^{i-1}\binom{k-1+i}{k}\equiv 0\pmod p.$

Poly-Bernoulli numbers are one of generalizations of the classical Bernoulli numbers. Since a negative index poly-Bernoulli number is an integer, it is an interesting problem to study this number from combinatorial viewpoint. In this short…

We consider $cp_{a,b,m}(n)$, the number of $(a,b,m)$-copartitions of $n$. We find many infinitelymany congruencesmodulo 2 and 6 for some particular value of $a$, $b$ and $m$.