Two different probability distributions are both known in the literature as "the" noncentral hypergeometric distribution. Wallenius' noncentral hypergeometric distribution can be described as an urn model with bias. Fisher's noncentral hypergeometric distribution is the conditional distribution of independent binomial variates given their sum. Several different methods for calculating probabili...

Two different probability distributions are both known in the literature as "the" noncentral hypergeometric distribution. The nomenclature problems are discussed. Wallenius' noncentral hypergeometric distribution can be described as an urn model with bias. Fisher's noncentral hypergeometric distribution is the conditional distribution of independent binomial variates given their sum. Several di...

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 ...

Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeomet-ric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E 7 (ellip-tic, hyperbolic) and of type E 6...

The transformation which assigns to a linear operator L the recurrence satis ed by coe cient se quences of the polynomial series in its kernel is shown to be an isomorphism of the corresponding operator algebras We use this fact to help factoring q di erence and recurrence operators and to nd nice power series solutions of linear di erential equations In particular we characterize generalized h...

The Schwarz map of the hypergeometric differential equation is studied first by Schwarz, and later by several authors for various generalizations of the hypergeometric equation. But up to now nothing is studied about the Schwarz map for confluent equations, mainly because such a map would produce just a chaos. Recently we defined the hyperbolic Schwarz map, and studied in several cases, includi...

The incomplete gamma function and incomplete beta function are important special functions in statistics. In modern theory of special functions, we regard hypergeometric functions as pairings of twisted cycles and twisted cocycles [1, Chap 2]. However, domains of integrations of these incomplete functions are not cycles. Solutions of A-hypergeometric systems ([3], [7, p.221]) have integral repr...

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 ...

We present an algorithm which, given a hypergeometric term T (n), constructs hypergeometric terms T1(n) and T2(n) such that T (n) = T1(n + 1) ? T1(n) + T2(n), and T2(n) is minimal in some sense. This solves the decomposition problem for indeenite sums of hypergeometric terms: T1(n + 1) ? T1(n) is the \summable part" and T2(n) the \non-summable part" of T (n).

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...