Related papers: On two congruences involving Franel numbers

We recall a proof of Euler's identity $\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$ involving the evaluation of a double integral. We extend the method to find Hurwitz Zeta series of the form $S(k,a)=\sum_{n \in \mathbb{Z}}…

We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These…

We calculate the least upper bounds of pointwise and uniform approximations for classes of $2\pi$-periodic functions expressible as convolutions of an arbitrary square summable kernel with functions, which belong to the unit ball of the…

The Ap\'ery polynomials are given by $$A_n(x)=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2x^k\ \ (n=0,1,2,\ldots).$$ (Those $A_n=A_n(1)$ are Ap\'ery numbers.) Let $p$ be an odd prime. We show that…

Z.-W. Sun introduced three kinds of numbers: \begin{align*}S_n=\sum_{k=0}^{n}{n\choose k}^2{2k\choose k}(2k+1),\qquad s_n=\sum_{k=0}^{n}{n\choose k}^2{2k\choose k}\frac{1}{2k-1}, \end{align*} and $S_n^{+}=\sum_{k=0}^{n}{n\choose…

The purpose of this note is to prove estimates for $$ \left| \sum_{k=1}^{n} \mbox{sign} \left( \cos \left( \frac{2\pi a}{n} k \right) \right) \mbox{sign} \left( \cos \left( \frac{2\pi b}{n} k \right) \right)\right|,$$ when $n$ is prime and…

In $1735$ Euler \cite{1} proved that for each positive integer $k$, the series $\zeta(2k) = \sum_{\ell=1}^{\infty} \ell^{-2k}$ converges to a rational multiple of $\pi^{2k}$. Many demonstrations of this fact are now known, and Euler's…

We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels…

In this paper, we prove two supercongruences by the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{(p-1)/2}\frac{3n+1}{(-8)^n}\binom{2n}n^3\equiv…

Consider an ordinary generating function $\sum_{k=0}^{\infty}c_kx^k$, 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…

We present some congruences modulo $p^{6-d}$ for sums of the type $\sum_{k=0}^{(p-3)/2}x^k{2k\choose k}/(2k+1)^d$, for $d=1,2,3$ where $p>5$ is a prime.

It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum_{\substack{i+j+k=p \\ i, j, k>0}} \frac{1}{i j k} \equiv-2 B_{p-3}\pmod p$$ has been…

We have gone back to old methods found in the historical part of Hardy's Divergent Series well before the invention of the modern analytic continuation to use formal manipulation of harmonic sums which produce some interesting formulae.…

Let $p$ be an odd prime, and let $a$ be a rational $p$-adic integer with $a\not\equiv 0\pmod p$. In this paper, using WZ method we establish the congruences for $\sum_{k=0}^{p-1} \binom ak^2(-1)^k(1-\frac 2ak)$ modulo $p^2$ and…

Binomial coefficients and harmonic numbers are important in many branches of number theory. With the help of the operator method and several summation and transformation formulas for hypergeometric series, we prove eight conjectural series…

In this paper, we prove a congruence which confirms a conjecture of Adamchuk. For any prime $p\equiv1\pmod3$ and $a\in\mathbb{Z}^{+}$, we have \begin{align*} \sum_{k=1}^{\frac{2}3(p^a-1)}\binom{2k}k\equiv0\pmod{p^2}. \end{align*}

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.

We prove that certain means of the quadratical partial sums of the two-dimensional Vilenkin-Fourier series are uniformly bounded operators from the Hardy space $H_{p}$ to the space $L_{p}$ for $0<p\leq 1.$ We also prove that the sequence in…

In this paper we study the exponential uniform strong summability of two-dimensional Vilenkin-Fourier series. In particular, it is proved that the two-dimensional Vilenkin-Fourier series of the continuous function $f$ is uniformly strong…

Asymptotic formulas for $\sum_{\kappa_j\le K}\alpha_j H_j^k(1/2)$ are proved when $k = 3,4$, where $H_j(s)$ is the Hecke series.