Related papers: Gauss sum with principal multiplicative character
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated…
In this article, we offer group-theoretic, field-theoretic, and topological interpretations of the Gaussian binomial coefficients and their sum. For a finite $p$-group $G$ of rank $n$, we show that the Gaussian binomial coefficient…
A subset of vertices of a graph $G$ is a general position set if no triple of vertices from the set lie on a common shortest path in $G$. In this paper we introduce the general position polynomial as $\sum_{i \geq 0} a_i x^i$, where $a_i$…
In the first part, we consider generalized quadratic Gauss sums as finite analogues of the Jacobi theta function, and the reciprocity law for Gauss sums as their transformation formula. We attach finite Dirichlet series to Gauss sums using…
A finite group is called $\psi$-divisible iff $\psi(H)|\psi(G)$ for any subgroup $H$ of a finite group $G$. Here, $\psi(G)$ is the sum of element orders of $G$. For now, the only known examples of such groups are the cyclic ones of…
We give summation formulae for the bilateral basic hypergeometric series ${}_1\psi_1( a; b; q, z )$ through Ramanujan's summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian…
This paper provides an estimate of the sum of a homogeneous polynomial $P$ of degree $\nu$ and mean zero over the lattice points inside a sphere of radius $R$. It is proved that $$ \sum_{\mathbf x \in \mathbb Z^3 \atop |\mathbf x| \le R}…
We present new ideas for computing elliptic Gau{\ss} sums, which constitute an analogue of the classical cyclotomic Gau{\ss} sums and whose use has been proposed in the context of counting points on elliptic curves and primality tests. By…
In this paper, generalizing the result in \cite{GXY}, we construct strongly regular Cayley graphs by using union of cyclotomic classes of $\F_q$ and Gauss sums of index $w$, where $w\geq 2$ is even. In particular, we obtain three infinite…
This paper develops new combinatorial approaches to analyze and compute special set partitions, called complementary set partitions, which are fundamental in the study of generalized cumulants. Moving away from traditional graph-based and…
The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}X_j^{2m_j}]\geq\prod_{j=1}^{n}E[X_j^{2m_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and $m_1,\dots,m_n\in\mathbb{N}$. In this…
In this paper, using combinatorial and analytic methods, we prove an exact calculating formula on the $2m$-th power mean value of the generalized quadratic Gauss sums for $m\geq 2$. This solves a conjecture of He and Zhang [`On the $2k$-th…
In this short note, we find an equivalent combinatorial condition only involving finite sums under which a centered Gaussian random vector with multinomial covariance matrix satisfies the Gaussian product inequality (GPI) conjecture. These…
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…
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…
In this paper, we study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in \cite{feng1}, where the Jacobi sums over Galois rings with characteristics a…
A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U = \alpha A + \beta I + \gamma J + \delta D$ with $\alpha \neq 0$, where $I$ is the identity matrix, $J$ is the all-ones matrix and $D$ is the…
In this paper we compute the sum of the $k$-th powers over any finite commutative unital rings, thus generalizing known results for finite fields, the rings of integers modulo $n$ or the ring of Gaussian integers modulo $n$. As an…
In the first chapter, we will present a computation of the square value of the module of L functions associated to a Dirichlet character. This computation suggests to ask if a certain ring of arithmetic multiplicative functions exists and…