On the Cauchy problem for a class of iterated Fuchsian partial differential equations

In this paper, we consider the Cauchy problem with ramified data for a class of iterated Fuchsian partial differential equations. We give an explicit representation of the solution in terms of Gauss hypergeometric functions. Our results are illustrated through some examples.     

负超几何随机变量期望与方差的一种简捷计算

给出了负超几何分布的概率模型,通过将负超几何分布随机变量进行和式分解,比较简捷地计算了它的期望和方差,并指出文献[4]计算的期望和方差是错误的.     

Sums of products of hypergeometric Bernoulli numbers

We give a formula for sums of products of hypergeometric Bernoulli numbers. This formula is proved by using special values of multiple analogues of hypergeometric zeta functions.     

Duality for hypergeometric functions and invariant Gauss-Manin systems

We present some basic identities for hypergeometric functions associatedwith the integrals of Euler type. We give a geometrical proof for formulaesuch as the identity between the single and double integrals expressingAppell's hypergeometric series F1 (a, b, b' c; x, y).     

Moments of Ramanujan?s generalized elliptic integrals and extensions of Catalan?s constant

We undertake a thorough investigation of the moments of Ramanujan?s alternative elliptic integrals and of related hypergeometric functions. Along the way we are able to give some surprising closed forms for Catalan-related constants and various new hypergeometric identities.     

On the better behaved version of the GKZ hypergeometric system

We consider a version of the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinsky (GKZ) suited for the case when the underlying lattice is replaced by a finitely generated abelian group. In contrast to the usual GKZ hypergeometric system, the rank of the better behaved GKZ hypergeometric system is always the expected one. We give largely self-contained proofs of many properties of this system. The discussion is intimately related to the study of the variations of Hodge structures of hypersurfaces in algebraic tori.     

On the order of starlikeness of the shifted Gauss hypergeometric function

We determine for several ranges of real parameters the order of starlikeness of the shifted Gauss hypergeometric function and we give some consequences of our results, in particular some mapping properties of the Carlson-Shaffer convolution operator.     

General multi-sum transformations and some implications

We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence \(\{g(k)\}\)), to be reduced to an infinite q-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some q orthogonal polynomials and various multi-sums that are expressible as infinite products.     

Hypergeometric Solutions of Some Algebraic Equations

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

Apostol-Bernoulli多项式的一个新公式

我们得到Apostol-Bernoulli多项式的一个用Gauss超几何函数表示的新公式,并给出了它的一些特殊情况和应用.     

Heun equations coming from geometry

We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pull-backs of hypergeometric functions by rational Belyi functions with restricted ramification data give rise to Heun equations.     

A brief account of Klein’s icosahedral extensions

We present an alternative relatively easy way to understand and determine the zeros of a quintic polynomial whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in his famous Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade are here shortened via Heymann’s theory of resolvents. Also, we give a complete explanation of the so-called icosahedral equation and its solution in terms of Gaussian hypergeometric functions. As an innovative element, we construct this solution by using algebraic transformations of hypergeometric series.     

Connection and inversion coefficients for basic hypergeometric polynomials

In this paper, we give a closed-form expression of the inversion and the connection coefficients for general basic hypergeometric polynomial sets using some known inverse relations. We derive expansion formulas corresponding to all the families within the q-Askey scheme and we connect some d-orthogonal basic hypergeometric polynomials.     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

A new approach to hypergeometric transformation formulas

The Ramanujan Journal - We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi’s canonical form of the...     

A hypergeometric approach,via linear forms involving logarithms,to criteria for irrationality of Euler’s constant

Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logarithms from Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.      

An Ansatz for the asymptotics of hypergeometric multisums

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singularities in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive K-theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.