The Hypergeometric Equation and Ramanujan Functions

In this paper we give analogues of the Ramanujan functions and nonlinear differential equations for them. Investigating a modular structure of solutions for nonlinear differential systems, we deduce new identities between the Ramanujan and hypergeometric functions. Another result of this paper is a solution of transcendence problems concerning nonlinear systems.     

Rational Hypergeometric Functions

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.     

p-adic超几何函数与Dwork超曲面上的有理点

p-adic超几何函数是经典的Gauss超几何函数在有限域上的模拟,与许多数论问题都有联系.设Fq是q元有限域,λ∈Fq,n为正整数.本文研究了Dwork超曲面Dλ^n:x1^n+x2^n+…+xn^n=nλx1x2…xn及其推广形式上的Fq-有理点,并在n与q(q-1)互素时给出了由p-adic超几何函数表示的各种Fq-有理点个数的公式,从而修正和改进了Barman与Goodson等人的结论.     

p-adic 超几何函数与 Dwork超曲面上的有理点

p-adic超几何函数是经典的Gauss超几何函数在有限域上的模拟,与许多数论问题都有联系.设F_q是q元有限域,λ∈F_q,n为正整数.本文研究了Dwork超曲面D_λⁿ：x₁ⁿ+x₂ⁿ+…+x_nⁿ=nλx₁x₂…x_n及其推广形式上的F_q-有理点,并在n与q（q-1）互素时给出了由p-adic超几何函数表示的各种F_q-有理点个数的公式,从而修正和改进了Barman与Goodson等人的结论.     

Properties of Some Families of Meromorphic Multivalent Functions Associated with Generalized Hypergeometric Functions

We introduce and study two subclasses ?_([α_1])(A, B, λ) and ?_([α_1])~+ (A, B, λ) of meromorphic p-valent functions defined by certain linear operator involving the generalized hypergeometric function. The main object is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses. We also derive many interesting results for the Hadamard products of functions belonging to the class ?_([α_1])~+(α, β, γ, λ).     

On Hypergeometric Functions and Generalizations of Legendre's Relation

We prove some convexity properties for a sum of hypergeometric functions and obtain a generalization of Legendre's relation for complete elliptic integrals. We apply these results to prove some inequalities for hypergeometric functions, incomplete beta-functions, and Legendre functions.     

Geometric Properties of Generalized Hypergeometric Functions

Let denote the generalized hypergeometric function where no denominator parameter can be zero or a negative integer and (a,n) denotes the ascending factorial notation. Ponnusamy and Vuorinen raised the problem of finding conditions on the parameters a_j > 0, b_j > 0 so that the function is univalent in . The main aim of this paper is to discuss this problem in detail for the case q = 2.     

Congruences for 2 F 1 Hypergeometric Functions Over Finite Fields

In a recent paper, Ono and Penniston proved a family of congruences for ₃ F ₂ hypergeometric functions over finite fields. They use the relationship between these functions and the arithmetic of a certain family of elliptic curves to obtain their congruences. Here we prove analogous congruences for ₂ F ₁ hypergeometric functions.     

Hypergeometric Functions Associated to Cones of Convex Type in a Euclidean Jordan Algebra

In this paper, we study a family of hypergeometric functions associated to cones of convex type in a Euclidean Jordan algebra. We obtain upper and lower bounds and give information about their singularities. The family under consideration was introduced by Sahi (Invent. Math.110 (1992), 409-418) and it plays an important role in the construction of some small representations.     

Computing the Real Zeros of Hypergeometric Functions

Efficient methods for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. These methods are based on global fixed point iterations which apply to families of functions satisfying first order linear difference differential equations with continuous coefficients. In order to compute the zeros of arbitrary solutions of the hypergeometric equations, we have at our disposal several different sets of difference differential equations (DDE). We analyze the behavior of these different sets regarding the rate of convergence of the associated fixed point iteration. It is shown how combinations of different sets of DDEs, depending on the range of parameters and the dependent variable, is able to produce efficient methods for the computation of zeros with a fairly uniform convergence rate for each zero.     

Hypergeometric Solutions of Some Algebraic Equations

We give the hypergeometric solutions of some algebraic equations including the general fifth-degree equation.     

Hypergeometric Solutions of Trigonometric KZ Equations Satisfy Dynamical Difference Equations

The trigonometric KZ equations associated to a Lie algebra g depend on a parameter λ∈h where h⊂g is a Cartan subalgebra. A system of dynamical difference equations with respect to λ compatible with the KZ equations is introduced by V. Tarasov and the second author (2000, Internat. Math. Res. Notices15, 801-829). We prove that the standard hypergeometric solutions of the trigonometric KZ equations associated to sl_N also satisfy the dynamical difference equations.     

An Expression for the Superposition of General Algebraic Functions in Terms of Hypergeometric Series

We show that the superposition of general algebraic functions is representable as a ratio of hypergeometric series.     

Hypergeometric Functions Related to Schur Q-Polynomials and the BKP Equation

We introduce hypergeometric functions related to projective Schur functions Q and describe their properties. Linear equations, integral representations, and Pfaffian representations are obtained. These hypergeometric functions are vacuum expectations of free fermion fields and are therefore tau functions of the so-called BKP hierarchy of integrable equations.     

Stable Recovery of Analytic Functions using Basic Hypergeometric Series

Two stable sampling formulas for reconstructing analytic functions from exponentially spaced samples are considered. Criterion for selecting regularization parameter and error estimates are obtained.     

Some Examples of RS 2 3(3)-Transformations of Ranks 5 and 6 as the Higher Order Transformations for the Hypergeometric Function

A combination of rational mappings and Schlesinger transformations for a matrix form of the hypergeometric equation is used to construct higher order transformations for the Gauss hypergeometric function.     

几个基本超几何级数公式

本文用 Bailey的变换公式和 Ismail等人的恒等式给出了一个新的 q-级数恒等式 .给出了这种方法的新的应用     

Multidimensional Matrix Inversions and Ar and Dr Basic Hypergeometric Series

We compute the inverse of a specific infinite r-dimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly, and Bhatnagar. Our inversion is an r-dimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of another infinite r-dimensional matrix. As applications of our matrix inversions, we derive new summation formulas for multidimensional basic hypergeometric series.     

A Heat Kernel Associated to Ramanujan's Tau Function

We prove a conjecture of Zagier, that the inverse Mellin transform of the symmetric square L-function attached to Ramanujan's tau function has an asymptotic expansion in terms of the zeros of the Riemann function.