Related papers: Aspects of elliptic hypergeometric functions
This paper explores the calculus of dual-valued functions and investigates the gamma function, beta function and generalized hypergeometric functions by incorporating dual numbers as parameters and variables. We examine its fundamental…
Eric Rains conjectured several quadratic transformations between multivariate elliptic hypergeometric functions in "Elliptic Littlewood Identities", with the integrand multiplied by interpolation functions. In this article two of these…
We give an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, we list a number of summations, transformations and explicit…
In this paper, we aim to present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella hypergeometric functions, by using the extended Bessel function due to Boudjelkha [4].…
In this paper, we prove two structural theorems on the general Berndt-type integrals with the denominator having arbitrary positive degrees by contour integrations involving hyperbolic and trigonometric functions, and hyperbolic sums…
Within the Ramanujan theories of elliptic functions, Li-Chien Shen constructed natural elliptic functions in signature three and signature four. When applied in signature six, the same constructions produce non-elliptic functions that…
We introduce a class of functions which constitutes an obvious elliptic generalization of multiple polylogarithms. A subset of these functions appears naturally in the \epsilon-expansion of the imaginary part of the two-loop massive sunrise…
The elliptic gamma function is a generalization of the Euler gamma function. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function. We prove multiplication formulas for the elliptic gamma…
We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure math- ematics and string theory. We then…
The authors survey recent results in special functions of classical analysis and geometric function theory, in particular the circular and hyperbolic functions, the gamma function, the elliptic integrals, the Gaussian hypergeometric…
We describe algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for…
The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or…
We develop the basic formulae of hyperspherical trigonometry in multidimensional Euclidean space, using multidimensional vector products, and their conversion to identities for elliptic functions. We show that the basic addition formulae…
We obtain some Bailey pairs associated with indefinite quadratic forms with the $\beta_n$ connected to a finite sum. A new general identity is given, which provides identities for $q$-hypergeometric series, including mock theta functions.
Present and future high-precision tests of the Standard Model and beyond for the fundamental constituents and interactions in Nature are demanding complex perturbative calculations involving multi-leg and multi-loop Feynman diagrams.…
Recently, many researchers devoted their attention to study the extensions of the gamma and beta functions. In the present work, we focus on investigating some approximations for a class of Gauss hypergeometric functions by exploiting…
An elliptic $BC_n$ generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter $BC_n$ Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root…
The aim of this article is to give a generalization of the Cauchy-Pompeiu integral formula for functions valued in parameter-depending elliptic algebras with structure polynomial $X^2 + \beta X + \alpha$ where $\alpha$ and $\beta$ are real…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
We establish a determinant formula for the bilinear form associated with the elliptic hypergeometric integrals of type $BC_n$ by studying the structure of $q$-difference equations to be satisfied by them. The determinant formula is proved…