Montgomery's identities for function of two variables

A New Class of Theta Function Identities in Two Variables

We describe a new series of identities, which hold for certain general theta series, in two completely independent variables. We provide explicit examples of these identities involving the Dedekind eta function, Jacobi theta functions, and various theta functions of Ramanujan. Introduction Let z ∈ H = {x+ yi : x, y ∈ R, y > 0} and for each x ∈ R set q = exp(2πixz) and e(x) = exp(2πix). The Dede...

Approximations for a Function of Two Variables

A discussion of the convergence of approximating functions for a given function of two variables ƒ(#, y), just as in the case of functions of one variable, can be based on two sets of theorems: (1) theorems on the existence of functions of closest approximation; (2) theorems on the representation of ƒ(#, y) by means of finite sums constructed in a specific way. From the first group we shall mak...

Estimation of the Survival Function for Negatively Dependent Random Variables

Let be a stationary sequence of pair wise negative quadrant dependent random variables with survival function {,1}nXn?F(x)=P[X>x]. The empirical survival function ()nFx based on 12,,...,nXXX is proposed as an estimator for ()nFx. Strong consistency and point wise as well as uniform of ()nFx are discussed

On Humbert matrix function of two complex variables under differential operator

Tail dependence for weighted mean of two copula function

In this paper, we study the properties of power weighted means, arithmetic, geometry and harmonic for two copulas.