相关论文: An arithmetic function of two variables

Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…

We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.

Additive relations are defined over additive monoids and additive operation is introduced over these new relations then we build algebraic system of equations. We can generate profuse equations by additive relations of two variables. To…

The purpose of this note is to provide an alternative proof of two quadratic transformation formulas contiguous to that of Gauss using a differential equation approach.

A conjecture concerning some pairs of interfering estimates for some integrals is formulated in three equivalent versions. Its importance for the the Paley problem for plurisubharmonic functions and for certain classes of extremal problems…

We consider double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the M\"obius function, and so on. We show analytic continuations of them by use of the Mellin-Barnes integral. Furthermore we…

Productions functions map the inputs of a firm or a productive system onto its outputs. This article expounds generalizations of the production function that include state variables, organizational structures and increasing returns to…

In this paper, we intend to revisit Theorem 2 of [3] formulating it in a way that, weakening the hypotheses and, at the same time, highlighting the richer conclusion allowed by the proof, it can potentially be applicable to a broader range…

An inequality is derived for the correlation of two univariate functions operating on symmetric bivariate normal random variables. The inequality is a simple consequence of the Cauchy-Schwarz inequality.

We find an arc-parameterization of the contour on which an given analytic function has constant modulus. This contour is seen to satisfy a differential equation which we explicitly give.

We give an explicit formula for the symbol of a function of an operator. Given an operator {\hat A} on L^2(R^N) with symbol A, and a function f, we obtain the symbol of f(\hat A) in terms of A. As an application, Bohr-Sommerfeld…

We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary…

We prove an inversion formula for summatory arithmetic functions. As an application, we obtain an arithmetic relationship between summatory Piltz divisor functions and a sum of the M\"obius function over certain integers, denoted by…

We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements of $\mathbb{F}_q[T]$ into primes and the factorizations of…

Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the…

In the paper, the authors establish, by using Cauchy integral formula in the theory of complex functions, an integral representation for the geometric mean of $n$ positive numbers. From this integral representation, the geometric mean is…

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(T^*(f(x),f(y)))$ where $T^*:[0,1]^2\rightarrow[0,1]$ is an associative function with neutral element in $[0,1]$, $f: [0,1]\rightarrow [0,1]$ is…

We investigate a linear operator associated with a functional equation that arises from studying some class of invariant measures under multidimensional transformations. By examining its iterates, we derive an explicit solution formula for…

Homogeneous and inhomogeneous biharmonic equation are considered on the $n$-dimensional unit sphere. The Green function is given as a series of Gegenbauer polynomials. In the paper, explicit representations of the Green function are found…

We describe an inequality of finite or infinite sequences of real numbers and their quotients. More precisely, we compare the quotient of H\"older functionals of two sequences of numbers with the sum of their quotients. In the last section…