In this paper we compute the Galois groups of basic hypergeometric equations. In this paper q is a complex number such that 0 < |q| < 1. 1 Basic hypergeometric series and equations The theory of hypergeometric functions and equations dates back at least as far as Gauss. It has long been and is still an integral part of the mathematical literature. In particular, the Galois theory of (generalize...

which has since been known as Kummer's theorem. This appears to be the simplest relation involving a hypergeometric function with argument ( — 1). All the relations in the theory of hypergeometric series rF8 which have analogues in the theory of basic series are those in which the argument is ( + 1 ) . Apparently, there has been no successful at tempt to establish the basic analogue of any form...

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler’s Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Ch...

In this course we will study multivariate hypergeometric functions in the sense of Gel’fand, Kapranov, and Zelevinsky (GKZ systems). These functions generalize the classical hypergeometric functions of Gauss, Horn, Appell, and Lauricella. We will emphasize the algebraic methods of Saito, Sturmfels, and Takayama to construct hypergeometric series and the connection with deformation techniques in...

In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those varieties we calculate values of Hypergeometric series on certain CM points. Our methods is based on the calculation of the Picard-Fuchs equations in higher dimen...

where a, b, c are rational parameters. By specialization of the parameters, Euler obtained the various classical functions that were around at that time. For example, taking b = c = 1 gives us Newton’s binomial series for (1 − z)−a and taking a = b = 1/2, c = 3/2 gives us arcsin(√z)/√z. Finally, taking all parameters equal to 1 recovers the ordinary geometric series, which more or less explains...