Exton introduced 20 distinct triple hypergeometric functions whose names are Xi (i = 1, . . . , 20) to investigate their twenty Laplace integral representations whose kernels include the confluent hypergeometric functions 0F1, 1F1, a Humbert function Ψ1, and a Humbert function Φ2. The object of this paper is to present 18 new integral representations of Euler type for the Exton hypergeometric f...

We give an introduction to the Gauss hypergeometric function, the hypergeometric equation and their properties in an elementary way. Moreover we explicitly and uniformly describe the connection coefficients, the reducibility of the equation and the monodromy group of the solutions.

The mathematical properties of the generalized hypergeometric function are now well established, due largely to the efforts of Bailey, Watson, Slater, and many others. There are available a number of books on the subject ['], [6] which contain extensive bibliographies together with an outline of the history of the investigations. A large portion of the published work concerning these functions ...

Grothendieck’s dessins d’enfants are applied to the theory of the sixth Painlevé and Gauss hypergeometric functions, two classical special functions of isomonodromy type. It is shown that higher-order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Bely̆ı functions. Moreover, deformations of the dessins d’enfants are introduced, ...

We survey several results on A-hypergeometric systems of linear partial differential equations introduced by Gelfand, Kapranov and Zelevinsky in the case of integer (and thus resonant) parameters, in particular, those differential systems related to sparse systems of polynomial equations. We also study in particular the case of A-hypergeometric systems for which kerA has rank 1. This allows us ...

Hypergeometric functions and their inequalities have found frequent applications in various fields of mathematical sciences. Motivated by the above, we set up certain including extended type Gauss hypergeometric function confluent function, respectively, virtue Hölder integral inequality Chebyshev’s inequality. We also studied monotonicity, log-concavity, log-convexity functions, which are deri...

Formulas and identities involving many well-known special functions (such as the Gamma and Beta functions, the Gauss hypergeometric function, and so on) play important rôles in themselves and in their diverse applications. Various families of generating functions have been established by a number of authors in many different ways. In this paper, we aim at establishing some (presumably new) gene...

The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences (1/2, 1/4, 1/4), (1/2, 1/3, 1/6) and (1/3, 1/3, 1/3). These form a special class of algebraic transformations of Gauss hypergeometric functions, of arbitrary high degree. The Gauss hypergeometric functions can be identified as elliptic integrals on the genus 1 curves y = x − x or...

We give a brief review of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler’s beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic ...