We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems. The basic hypergeometric function in question was constructed as explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted affine root system is reduced. Its construction was extended to the nonreduced case by the author. It is a meromorphic We...

Hypergeometric sequences are such that the quotient of two successive terms is a xed rational function of the index. We give a generalization of M. Petkov sek's algorithm to nd all hypergeometric sequence solutions of linear recurrences, and we describe a program to nd all hypergeometric functions that solve a linear diierential equation. Solutions hyperg eom etriques des equations dii erentiel...

A weight-dependent generalization of the binomial theorem for noncommuting variables is presented. This result extends the well-known binomial theorem for q-commuting variables by a generic weight function depending on two integers. For two special cases of the weight function, in both cases restricting it to depend only on a single integer, the noncommutative binomial theorem involves an expan...

The bivariate confluent hypergeometric function kind 1 distribution is defined by the probability density function proportional to x1 1 x2 2 1 F1(α; β; −x1 − x2). In this article, we study several properties of this distribution and derive density functions of X1/X2, X1/(X1 + X2), X1 + X2 and 2 √ X1X2. The density function of 2 √ X1X2 is represented in terms of modified Bessel function of the s...

A bivariate gamma-type density function involving a confluent hypergeometric function of two variables is being introduced. The inverse Mellin transform technique is employed in conjunction with the transformation of variable technique to obtain its moment generating function, which is expressed in terms of generalized hypergeometric functions. Its cumulative distribution function is given in c...

Abstract. We find two-sided inequalities for the generalized hypergeometric function with positive parameters restricted by certain additional conditions. Our lower bounds are asymptotically precise at x = 0, while the upper bounds are either asymptotically precise or at least agree with q+1Fq((aq+1), (bq);−x) at x = ∞. Inequalities are derived as corollaries of a theorem asserting the monotony...

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 ...

We provide a short and elementary proof of the main technical result recent article “Uniqueness Clifford torus with prescribed isoperimetric ratio” by Thomas Yu Jingmin Chen [Proc. Amer. Math. Soc. 150 (2022), pp. 1749–1765]. The key new is an explicit expression central function ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-...

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...

for orbitals ψ centered at places A, B, C and D. The Gauss Transformation Method has been shown to calculate the integral if the orbitals ψ are expanded in a basis of Gaussian Type Orbitals (GTO’s) [4]; this manuscript basically demonstrates how dealing with the quadratic forms in the exponentials directly also manages to reduce them to the omnipresent Confluent Hypergeometric Functions of the ...