Related papers: Some congruences involving generalized Bernoulli n…

We derive two new generalizations of the Busche-Ramanujan identities involving the multiple Dirichlet convolution of arithmetic functions of several variables. The proofs use formal multiple Dirichlet series and properties of symmetric…

Let $M=M_1 M_2 M_3$ be the product of three distinct primes and let $\chi=\chi_1 \chi_2 \chi_3$ be a Dirichlet character of modulus $M$ such that each $\chi_i$ is a primitive character modulo $M_i$ for $i=1,2,3$. In this paper, we provide a…

We introduce poly-Bernoulli polynomials in two variables by using a generalization of Stirling numbers of the second kind that we studied in a previous work. We prove the bi-variate poly-Bernoulli polynomial version of some known results on…

In this paper we use probabilistic methods to derive some results on the generalized Bernoulli and generalized Euler polynomials. Our approach is based on the properties of Appell polynomials associated with uniformly distributed and…

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a couple of infinite families of explicit formulas for $S_k(n)$. One of the families…

We establish two binomial coefficient--generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of…

In this paper our main theorem states the following, Main Theorem : Let B denote the polynomial ring D[x1,.... ,xn] , in the commuting indeterminates x i over a division ring D . Let M be a finitely generated B-module . Let B m denote the…

In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence…

Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, Lehmer's generalized Euler numbers are studied to give certain…

Let $B_{l,m}(n)$ denote the number of $(l,m)$-regular bipartitions of $n$. Recently, many authors proved several infinite families of congruences modulo $3$, $5$ and $11$ for $B_{l,m}(n)$. In this paper, using theta function identities to…

Let $q\ge3$ be an integer, $\chi$ be a Dirichlet character modulo $q$, and $L(s,\chi)$ denote the Dirichlet $L$-functions corresponding to $\chi$. In this paper, we show some special function series, and give some new identities for the…

In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine…

Let $N_k(n,r,\boldsymbol{a})$ denote the number of incongruent solutions of the quadratic congruence $a_1x_1^2+\ldots+a_kx_k^2\equiv n$ (mod $r$), where $\boldsymbol{a}=(a_1,\ldots,a_k)\in {\Bbb Z}^k$, $n\in {\Bbb Z}$, $r\in {\Bbb N}$. We…

Let $(r_{A,n}(x))_{n \in \mathbb{N}}$ be a sequence of polynomials with coefficients from a field $K$ satisfying the recurrence relation $r_{A,n}(x)= \sum_{|\alpha|\leq m} t_{\alpha,n}(x)\textbf{r}_{A,n}^\alpha(x)$ of order $d+1 \in…

In this paper, we show that certain sums of generalized $m$-gonal numbers represent every positive integer if and only if they represent every positive integer up to an explicit bound $C_m$, verifying a conjecture of Sun for sufficiently…

We begin with the observation that the signed generalized Stirling polynomials $P_k(m,x)$, which occur in a generalization of Malmsten's integral, reduce to the falling factorials when $k=m$. The structure of these generalized Stirling…

In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers $B_{2 n}$, $n$ being any positive integer. This new recurrence seems advantageous in comparison to other known formulae since…

We provide direct elementary proofs of several explicit expressions for Bernoulli numbers and Bernoulli polynomials. As a byproduct of our method of proof, we provide natural definitions for generalized Bernoulli numbers and polynomials of…

We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$)…

We prove some polynomial identities from which we deduce congruences modulo $p^2$ for the Fermat quotient $\frac{2^p-2}{p}$ for any odd prime $p$ (Proposition 1 and Theorem 1). These congruences are simpler than the one obtained by…