Related papers: An identity for the Kloosterman sum

Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24], we give simple proofs of Jensen's identity $$ \sum_{k=0}^{n}{x+kz\choose k}{y-kz\choose n-k} =\sum_{k=0}^{n}{x+y-k\choose n-k}z^k, $$ and Chu's and Mohanty-Handa's…

We give a new simpler proof of a theorem of Jayne and Rogers.

Menon's identity is $\sum_{a \in A}^m (a-1,m) = d(m) \varphi(m)$, where $A$ is a reduced set of residues modulo $m$. This paper contains elementary proofs of some generalizations of this result.

In this paper we give combinatorial proofs of some well known identities and obtain some generalizations. We give a visual proof of a result of Chapman and Costas-Santos regarding the determinant of sum of matrices. Also we find a new…

We give generating functions for Gauss sums for finite general linear and unitary groups. For the general linear case only our method of proof is new, but we deduce a bound on Kloosterman sums which is sometimes sharper than Deligne's bound…

We use some elementary arguments to obtain a new bound on bilinear sums with weighted Kloosterman sums which complements those recently obtained by E. Kowalski, P. Michel and W. Sawin (2020).

For W a finite Coxeter group, a formula is found for the size of W equivalence classes of subsets of a base. The proof is a case-by-case analysis using results and tables of Carter and Orlik/Solomon. As a corollary we obtain an alternating…

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.

Modifying an idea of E. Brietzke we give simple proofs for the recurrence relations of some sequences of binomial sums which have previously been obtained by other more complicated methods.

We give a new proof of Chan's identity involving the cubic partition function and we also give a new identity for the cubic partition function which is analogues to the Zuckerman's identity for the ordinary partition function.

This paper introduces a variation on an identity by Bruckman and Good. Using this identity, we are able to derive various well-known sums involving reciprocals of Fibonacci and Lucas numbers, including the case when the indices form an…

Based on Jensen formulae and the second kind of Chebyshev polynomials, another proof is presented for an extension of a curious binomial identity due to Z. W. Sun and K. J. Wu.

We present in this work a new and simple proof of the false centre theorem.

I revisit an automated proof of Andrews' pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews' result along with a variety of new…

Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of…

We prove that the Kloosterman sum $\text{Kl}(1,q)$ changes sign infinitely many times, as $q\rightarrow +\infty$ with at most six prime factors. As a consequence, our result improved the best known result of Xi(IMRN, 2022). The novelty of…

We systematically exploit a new generalized hypergeometric identity to obtain new hypergeometric summation formulas. As a consistency test, alternative proofs for some special cases are also provided. As a byproduct new summation formulas…

We define the Kloosterman sum for $SL_4$ over the Kloosterman set via the Bruhat decomposition and stratify the Kloosterman set using the reduced word decomposition of the Weyl group element. The Kloosterman sum for an $SL_4$-long word is…

In this note, we combine ideas of several previous proofs in order to obtain a quite short proof of Gr\"otzsch theorem.

In this short note we provide some algebraic identity with a proof exploiting its probabilistic interpretation. We show several consequences of the identity, in particular we obtain a new representation of a Stirling number of second kind,…