Related papers: Algebraicity of hypergeometric functions with arbi…

Universal algebraic geometry allows considering of geometric properties of every universal algebra. When two algebras have same algebraic geometry? We must consider the categories of algebraic closed sets of these algebras to answer this…

Beukers and Heckman gave necessary and sufficient conditions for a hypergeometric function $_n F_{n-1}$ to be algebraic. We give a new proof of this theorem by passing through the Mehta-Seshadri correspondence. In particular, we explicitly…

We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for…

This paper continues the work which attempts to understand the general properties of the graded algebras associated with Hecke symmetries without a restriction on the parameter q of the Hecke relation imposed in earlier results.

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two…

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…

We consider a certain definite integral involving the product of two classical hypergeometric functions having complicated arguments. We show in this paper the surprising fact that this integral does not depend on the parameters of the…

General elliptic hypergeometric functions are defined by elliptic hypergeometric integrals. They comprise the elliptic beta integral, elliptic analogues of the Euler-Gauss hypergeometric function and Selberg integral, as well as elliptic…

Identities involving finite sums of products of hypergeometric functions and their duals have been studied since 1930s. Recently Beukers and Jouhet have used an algebraic approach to derive a very general family of duality relations. In…

Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study…

A unifying scheme of classical special functions of hypergeometric type obeying orthogonality or biorthogonality relations is described. It expands the Askey scheme of classical orthogonal polynomials and its $q$-analogue based on the…

Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular,…

In this paper we consider basic hypergeometric functions introduced by Heine. We study mapping properties of certain ratios of basic hypergeometric functions having shifted parameters and show that they map the domains of analyticity onto…

In this article, we prove a generalized Rodrigues formula for a wide class of holonomic Laurent series, which yields a new linear independence criterion concerning their values at algebraic points. This generalization yields a new…

This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…

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…

We show that the hypergeometric groups associated to the pairs of the parameters $\left(0,0,\frac{1}{3}, \frac{2}{3}\right)$, $\left(\frac{1}{2},\frac{1}{2},\frac{1}{4},\frac{3}{4}\right)$; and $\left(0,\frac{1}{12},…

We develop a theoretical study of non-terminating hypergeometric summations with one free parameter. Composing various methods in complex and asymptotic analysis, geometry and arithmetic of certain transcendental curves and rational…

We present a criterion for the existence of telescopers for mixed hypergeometric terms, which is based on multiplicative and additive decompositions. The criterion enables us to determine the termination of Zeilberger's algorithms for mixed…

We present a systematic method for proving nonterminating basic hypergeometric identities. Assume that $k$ is the summation index. By setting a parameter $x$ to $xq^n$, we may find a recurrence relation of the summation by using the…