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"Hypergeometric functions"
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Hypergeometric functions over finite fields
Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we
consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions
over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the
classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric
transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As
an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation
formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain
hypergeometric varieties.
eBook
Bounded Littlewood identities
We describe a method, based on the theory of Macdonald–Koornwinder polynomials, for proving bounded Littlewood identities. Our
approach provides an alternative to Macdonald’s partial fraction technique and results in the first examples of bounded Littlewood
identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type
eBook
Quadratic transformation inequalities for Gaussian hypergeometric function
In the article, we present several quadratic transformation inequalities for Gaussian hypergeometric function and find the analogs of duplication inequalities for the generalized Grötzsch ring function.
Journal Article
3F4 Hypergeometric Functions as a Sum of a Product of 1F2 Functions
This paper shows that certain F43 hypergeometric functions can be expanded in sums of pair products of F21 functions. In special cases, the F43 hypergeometric functions reduce to F32 functions. Further special cases allow one to reduce the F32 functions to F21 functions, and the sums to products of F10 (Bessel) and F21 functions. The class of hypergeometric functions with summation theorems are thereby expanded beyond those expressible as pair-products of F12 functions, F23 functions, and generalized Whittaker functions, into the realm of Fqp functions where p
Journal Article
Numerical methods for the computation of the confluent and Gauss hypergeometric functions
The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss–Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide “roadmaps” with our recommendation for which methods should be used in each situation.
Journal Article
Bi-Concave Functions Connected with the Combination of the Binomial Series and the Confluent Hypergeometric Function
In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients |a2| and |a3| for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper.
Journal Article
Generalized Summation Formulas for the Kampé de Fériet Function
By employing two well-known Euler’s transformations for the hypergeometric function 2F1, Liu and Wang established numerous general transformation and reduction formulas for the Kampé de Fériet function and deduced many new summation formulas for the Kampé de Fériet function with the aid of classical summation theorems for the 2F1 due to Kummer, Gauss and Bailey. Here, by making a fundamental use of the above-mentioned reduction formulas, we aim to establish 32 general summation formulas for the Kampé de Fériet function with the help of generalizations of the above-referred summation formulas for the 2F1 due to Kummer, Gauss and Bailey. Relevant connections of some particular cases of our main identities, among numerous ones, with those known formulas are explicitly indicated.
Journal Article
Self-Similar Solutions of a Multidimensional Degenerate Partial Differential Equation of the Third Order
When studying the boundary value problems’ solvability for some partial differential equations encountered in applied mathematics, we frequently need to create systems of partial differential equations and explicitly construct linearly independent solutions explicitly for these systems. Hypergeometric functions frequently serve as solutions that satisfy these systems. In this study, we develop self-similar solutions for a third-order multidimensional degenerate partial differential equation. These solutions are represented using a generalized confluent Kampé de Fériet hypergeometric function of the third order.
Journal Article
Euler-type integral representations for the Kampé de Fériet functions
In this paper, the Kampé de Fériet functions of arbitrary orders and their Euler-type integral representations are studied. The general form of the integral representations for a Kampé de Fériet function are proved. Conditions, under which these representations are expressed in terms of products of two gen-eralized hypergeometric functions, are found. Examples are identified in which the integral representation of the Kampé de Fériet function contains an elementary function or a known second-order hypergeometric function of two variables.
Journal Article