Related papers: Wilson's theorem modulo p^2 derived from Faulhaber…

Updated version of 2013 Arizona WInter School notes on modularity lifting theorems for for two-dimensional p-adic representations, using wherever possible arguments that go over to the n-dimensional (self-dual) case.

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 the research, with aid of the Fa\`a di Bruno formula, be virtue of several identities for the Bell polynomials of the second kind, with help of two combinatorial identities, by means of the (logarithmically) complete monotonicity of…

We consider the numbers $\mathcal{B}_{r,s} = (\mathbf{B}+1)^r \mathbf{B}^s$ (in umbral notation $\mathbf{B}^n = \mathbf{B}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the…

Faulhaber's formula expresses the sum of the first $n$ positive integers, each raised to an integer power $p\geq 0$, as a polynomial in $n$ of degree $p+1$. Ramanujan expressed this sum for $p\in\{\frac12,\frac32,\frac52,\frac72\}$ as the…

Wolstenholme's type summations involve certain powers of all residues $k$ modulo some prime number $p$. We first consider the sums of double or triple products of certain powers of all residues, e.g., the sums of the terms $(a+k)^m(b+k)^n$…

Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)\cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ \sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3} \equiv…

In this note, the first-order Dickson polynomials are introduced through a particular case of the expression of the trace of the $n^{th}$ power of a matrix in terms of powers of the trace and determinant of the matrix itself. The technique…

In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler…

Sum of powers 1^p+...+n^p, with n and p being natural numbers and n>=1, can be expressed as a polynomial function of n of degree p+1. Such representations are often called Faulhaber formulae. A simple recursive algorithm for computing…

In this paper, we consider the degenerate poly-Bernoulli polynomials and present new and explicit formulas for computing them in terms of the degenerate Bernoulli polynomials and Stirling numbers of the second kind.

For integer $k \geq 0$, let $S_k$ denote the sum of the $k$th powers of the first $n$ positive integers $1^k + 2^k + \cdots + n^k$. For any given $k$, the power sum $S_k$ can in principle be determined by differentiating $k$ times (with…

We present a method for obtaining congruences modulo powers of a prime number~$p$ for combinatorial sequences whose generating function satisfies an algebraic differential equation. This method generalises the one by Kauers and the authors…

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…

We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and…

In the paper, by establishing a new and explicit formula for computing the $n$-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind…

We study congruences for Eisenstein series on $\mathrm{SL}_2(\mathbb{Z})$ modulo $p^2$, where $p \geq 5$ is prime. It is classically known that all Eisenstein series of weight at least $4$ are determined modulo $p^2$ by those of weight at…

In this paper we prove that for any prime $p\ge 11$ holds $$ {2p-1\choose p-1}\equiv 1 -2p \sum_{k=1}^{p-1}\frac{1}{k} +4p^2\sum_{1\le i<j\le p-1}\frac{1}{ij}\pmod{p^7}. $$ This is a generalization of the famous Wolstenholme's theorem which…

The aim of this paper is to construct generating functions for some families of special finite sums with the aid of the Newton-Mercator series, hypergeometric series, and $p$-adic integral (the Volkenborn integral). By using these…

In most text books on number theory Wilson Theorem is proved by applying Lagrange theorem concerning polynomial congruences.Hardy and Wright also give a proof using cuadratic residues. In this article Wilson theorem is derived as a…