Related papers: An identity for the Kloosterman sum

Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we…

We investigate the solubility of the congruence xy=1 (mod p), where p is a prime and x,y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.

Arguably the simplest variation of this style of proof as we avoid reducing to the cubic case entirely.

Let $q$ be a positive integer, $\chi$ a nontrivial character mod $q$, $\mathcal{I}$ an interval of length not exceeding $q.$ In this paper we shall study the character sum analogue of the well-known Kloosterman…

A recurrence relations for sums of powers of complex functions can be written as a system of linear equation AX=B. Using properties of determinant and Cramer's rule for solving systems of linear equation, this paper presents an absolutely…

We give a proof of two identities involving binomial sums at infinity conjectured by Z-W Sun. In order to prove these identities, we use a recently presented method i.e. we view the series as specializations of generating series and derive…

We present in this work a complete session in a Mathematica notebook. The aim of this notebook is to check identities in symmetric compositions. This notebook is a complement of our work [1] and it has all the explicit computations. We…

In this note, we give a simple method for computing the column sums of the Sonnenschein summability matrices.

Using the expansion in a Fourier-Gegenbauer series, we prove several identities that extend and generalize known results. In particular, it is proved among other results, that \begin{equation*}…

We present identities for permutations with fixed points. The formulas are based on successive derivations or integrations of the determinant of a particular matrix.

We prove upper bounds for sums of Kloosterman sums against general arithmetic weight functions. In particular, we obtain power cancellation in sums of Kloosterman sums over arithmetic progressions, which is of square-root strength in any…

In this short note, we give a new Menon-type identity involving the sum of element orders and the sum of cyclic subgroup orders of a finite group. It is based on applying the weighted form of Burnside's lemma to a natural group action.

We use an integral method to establish a number of Rogers-Ramanujan type identities involving double and triple sums. The key step for proving such identities is to find some infinite products whose integrals over suitable contours are…

We obtain several estimates for trilinear form with double Kloosterman sums. In particular, these bounds show the existence of nontrivial cancellations between such sums.

We present a new, elementary, dynamical proof of the prime number theorem.

We provide bijective proofs of two classic identities that are very simple to prove using generating functions, but surprisingly difficult to prove combinatorially. The problem of finding a bijective proof for the first identity was first…

For any homogeneous identity between $q$-minors, we provide an identity between $P,Q$-minors.

This paper presents both a proof method and a result. The proof method presented is particularly suitable for uniformly proving families of identities satisfied by a family of recursive sequences. To illustrate the method, we study the…

We give a Laurent series proof of the Habsieger-Kadell $q$-Morris identity, which is a common generalization of the $q$-Morris identity and the Aomoto constant term identity. The proof allows us to extend the theorem for some additional…

By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…