It is known that the radial equation of the massless fields with spin around Kerr black holes cannot be solved by special functions. Recently, the analytic solution was obtained by use of the expansion in terms of the special functions and various astrophysical application have been discussed. It was pointed out that the coefficients of the expansion by the confluent hypergeometric functions ar...

We present a fast and exact novel algorithm to compute maximum likelihood estimates for the number of defects initially contained in a software, using the hypergeometric software reliability model. The algorithm is based on a rigorous and comprehensive mathematical analysis of the growth behavior of the likelihood function for the hypergeometric model. We also study a numerical example taken fr...

We construct an explicit orthonormal basis of piecewise i+1Fi hypergeometric polynomials for the Alpert multiresolution analysis. The Fourier transform of each basis function is written in terms of 2F3 hypergeometric functions. Moreover, the entries in the matrix equation connecting the wavelets with the scaling functions are shown to be balanced 4F3 hypergeometric functions evaluated at 1, whi...

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...

Two different probability distributions are both known in the literature as “the” noncentral hypergeometric distribution. These two distributions will be called Fisher’s and Wallenius’ noncentral hypergeometric distribution, respectively. Both distributions can be associated with the classical experiment of taking colored balls at random from an urn without replacement. If the experiment is unb...

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 series with noncommutative parameters and argument, in the special case involving square matrices, have recently been studied by a number of researchers including (in alphabetical order) Durán, Duval, Grünbaum, Iliev, Ovsienko, Pacharoni, Tirao, and others. See [3, 6, 8, 9, 10, 11, 12, 16] for some selected papers. The subject of hypergeometric series involving matrices is closel...

In this work we derive a new integral of the hypergeometric differential equation, valid for arbitrary parameters α, β, γ and which is expressed in terms of indefinite integrals. This leads to new types of relations between F (α, β, γ;x) and its contiguous functions : F (α, β, γ;x) can be related to only one of the functions contiguous to it.These equations enable one to extend the formulae giv...

The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Clausenian hypergeometric 3F2 functions and polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Clausenian hypergeometric polynomials: 3F2(−n, τn+ a, b; τn+ c,−n+ d; z), where n is a nonnegative integer. Some families o...

the probability density functions fitting to the discrete probability functions has always been needed, and very important. this paper is fitting the continuous curves which are probability density functions to the binomial probability functions, negative binomial geometrics, poisson and hypergeometric. the main key in these fittings is the use of the derivative concept and common differential ...