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...

Let p be an odd prime. In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have been of interest as they are related to the number of Fp points on algebraic varieties and to Fourier coefficients of modular forms. In this paper, we explicitly determine these functions modulo higher powers of p and discuss an application to super...

where aq = bcdef (cf. [9, Eq. (2.11.7)]), is one of the deepest results in the classical theory of basic hypergeometric series. It contains many important identities as special cases (such as the nonterminating 3φ2 summation, the terminating 8φ7 summation, and all their specializations including the q-binomial theorem). One way to derive (1.1) is to start with a particular rational function ide...

The 15 Gauss contiguous relations for 2F1 hypergeometric series imply that any three 2F1 series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We prove several properties of coefficients of these general contiguous relations, and use the results to propose effective ways to compute contiguous relations. We also di...

Let p be prime and let GF (p) be the finite field with p elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions 2F1(x) =2 F1 „ φ, φ | x « and 3F2(x) =3 F2 „ φ, φ, φ , | x « where φ and respectively are the quadratic and trivial characters of GF (p). For all but finitely many rational numbers x = λ, there exist two elliptic curves 2E1(λ) and 3E2(...

Let $A$ be a set of $N$ vectors in ${\mathbb Z}^n$ and let $v$ vector C}^N$ that has minimal negative support for $A$. Such gives rise to formal series solution the $A$-hypergeometric system with parameter $\beta=Av$. If lies Q}^n$, then this rational coefficients. $p$ prime number. We characterize those whose coordinates are rational, $p$-integral, lie closed interval $[-1,0]$ which correspond...

In the preprint [1] one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and was not proved from 1993. In this paper we prove some conjectures from [1] for Kummer hypergeometric functions and its further generalizations for Gauss and generalized ...

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...