On Powers of a Hypergeometric Function
Michael Milgram
总结了不完全β函数的一种特殊情况下超几何函数的一种新的整幂展开式，并给出了两种新的双伽玛（psi）函数和的结果.
A new expansion for integral powers of the hypergeometric function corresponding to a special case of the incomplete beta function is summarized, and consequences, including two new sums involving digamma (psi) functions are presented.
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It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions have been used widely in various fields of applied mathematics and natural sciences. In this expository note, we provide a simple proof of the differentiation identities, which is based only on the definition of the coefficients for the power series expansion of the hypergeometric functions.
The second type of power mean bounds for the hypergeometric mean
本文利用特殊幂级数符号法则与拉普拉斯积分，得到超几何函数的最优幂均值界，其右端点处精确，所得结果涵盖诸多已知结论。
Some decomposition formulas of generalized hypergeometric functions and formulas of an analytic continuation of the Clausen function
本文利用Burchnall和Chaundy的相似符号方法构造了广义超几何函数的展开式。利用所得到的展开式，定义了克劳森超几何函数的解析延拓公式。所得到的解析延拓公式表示已知的超几何Appell函数$ F_2 ({a;b_1, b_2 ;c_1, c_2 ;x,y}) $，其理论已得到很好的研究。
Integral transformations of hypergeometric functions with several variables
本文推广Riemann-Liouville积分，引入收敛幂级数的积分变换，并应用于多变量超几何函数。
Essays on the theory of elliptic hypergeometric functions
本文对数学物理中一类新的特殊函数--椭圆超几何函数理论的主要结果作了简要的回顾。证明了推广欧拉β积分的最一般的一元精确积分公式，称为椭圆β积分。本文构造了高斯超几何函数的一个椭圆模拟及其椭圆超几何方程，并描述了该函数及其特殊子情形的双正交关系。我们列出已知的椭圆β积分的根系，并考虑对称变换为相应的椭圆超几何函数的高阶。
Aspects of elliptic hypergeometric functions
一般椭圆超几何函数由椭圆超几何积分定义。它们包括椭圆β积分，椭圆类似物的欧拉-高斯超几何函数和Selberg积分，以及椭圆扩展的许多其他平原超几何和$q$ -超几何建设。特别是，贝利链技术，用于证明罗杰斯-拉马努金型身份证，已被推广到积分。在椭圆的水平上，它产生一个解决方案的杨-巴克斯特方程作为一个积分算子与椭圆超几何内核。我们简要介绍了这一领域的发展情况。
HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4
HYPERDIRE是一个致力于创建一套基于Mathematica的程序的项目，用于超几何函数的微分约简。目前的版本包括两个部分：一个是pfq，用于操作超几何functions_{p+1} F_p;另一个是Appell F_1F_4，用于操作二元Appell超几何函数F_1，F_2，F_3，F_4。
General reduction of the elliptic hypergeometric equation to the level of complex hypergeometric functions is described. The derived equation is generalized to the Hamiltonian eigenvalue problem for new rational integrable $N$-body systems emerging from particular degenerations of the elliptic Ruijsenaars and van Diejen models.
Incomplete A-Hypergeometric Systems
不完全贝塔函数是统计学中一个重要的特殊函数。在现代超几何函数理论中，我们把超几何函数看成是扭圈与扭上圈的配对。然而，不完全的beta函数在这个方案中不能被理解;换句话说，积分的域不是循环（不完全）。我们将推广不完全函数的A-超几何系统的理论。对不完全高斯超几何函数进行了一般性研究和详细研究。
Integral representation and sharp asymptotic results for some Heckman-Opdam hypergeometric functions of type BC
BC型Heckman-Opdam超几何函数是经典Jacobi函数的单变量推广，它包含了实数、复数或四元数上的非紧Grassmann流形的球函数.对于这样的超几何函数，存在各种已知的极限转变。本文利用Harish-Chandra积分表示的一种显式形式和一种内插变量，得到了三类连续的BC型超几何函数的极限结果，它们分别具有显式的、尖锐的和一致误差界.第一个极限实现了固定秩的无限维格拉斯曼函数的球面函数的近似;这里A型的超几何函数作为极限出现。第二个极限是对Dunkl型Bessel函数的压缩极限。
We show that the complex hypergeometric function describing $6j$-symbols for $SL(2,\mathbb{C})$ group is a special degeneration of the $V$-function -- an elliptic analogue of the Euler-Gauss $_2F_1$ hypergeometric function. For this function, we derive mixed difference-recurrence relations as limiting forms of the elliptic hypergeometric equation and some symmetry transformations. At the intermediate steps of computations, there emerge a function describing the $6j$-symbols for the Faddeev modular double and the corresponding difference equations and symmetry transformations.
Elliptic Hypergeometric Functions
在这些课堂讲稿中，我将对椭圆超几何函数进行初步的介绍。我专注于激发主要思想和结构，而不是给出一个全面的调查。讲座包括在可解晶格模型的背景下椭圆超几何函数的历史起源的简要解释。特别地，我给出了一个新的证明的事实，即融合玻尔兹曼重量的椭圆固体上固体模型可以表示为椭圆超几何和。
On singular univariate specializations of bivariate hypergeometric functions
通过直接使用2F 1（1）的高斯公式计算x=0处的表观幂级数，可以很容易地计算F2（x，1）和Appell函数的类似单变量特化。但是这种简单的计算可能会导致错误，因为2F 1（1）系数最终可能会发散;然后x=0处的实际幂级数可能会涉及分支项。本文以具体实例说明了这些并发症。
Bernoulli inequality and hypergeometric functions
给出了对数型函数的Bernoulli型不等式。这些函数尤其包括零平衡情况下的高斯超几何函数$F(a,b;a+b;x)\,.$
On the exterior power structure of the cohomology groups for the general hypergeometric integral
本文研究 Gelfand 超几何函数及其合流族对应的代数 de Rham 上同调群，证明在参数位于 Veronese 像时，该群可分解为一维积分情形上同调群的外幂积。
We review the series solutions of the general and single-confluent Heun equations in terms of powers, ordinary-hypergeometric and confluent-hypergeometric functions. The conditions under which the expansions reduce to finite sums as well as the cases when the coefficients of power-series expansions obey two-term recurrence relations are discussed. Infinitely many cases for which a solution of a general or single-confluent Heun equation is written through a single generalized hypergeometric function are indicated. It is shown that the parameters of this generalized hypergeometric function are t…
We aim to introduce a new extension of beta function and to study its important properties. Using this definition, we introduce and investigate new extended hypergeometric and confluent hypergeometric functions. Further, some hybrid representations of this extended beta function are derived which include some well known special functions and polynomials.
In this work we show that it is possible to calculate the fractional integrals and derivatives of order $α$ (using the Riemann-Liouville formulation) of power functions $\left( t-\ast\right) ^β$ with $β$ being any real value, so long as one pays attention to the proper choosing of the lower and upper limits according to the original function's domain. We, therefore, obtain valid expressions that are described in terms of function series of the type $\left( t-\ast\right) ^{\pmα+k}$ and we also show that they are related to the famous hypergeometric functions of the Mathematical-Physics.
As a contribution to the Ramanujan theory of elliptic functions to alternative bases, Li-Chien Shen has shown how analogues of the Jacobian elliptic functions may be derived from incomplete hypergeometric integrals in signatures three and four. We determine precisely the signatures in which the Jacobian analogues or their squares are indeed elliptic.
An Inequality for Basic Confluent Hypergeometric Series
在这篇很短的注记中，我们将得到一类包含所有合流基本超几何级数的整函数的不等式和一类包含θ函数的亚纯函数的不等式.
On transformations of A-hypergeometric functions
本文系统地研究了$A$ -超几何函数的变换。我们的方法是应用变量的变化对应的自同构的环面环，欧拉型整表示的$A$ -超几何函数。我们证明了所有的线性$A$ -超几何变换都是由相应多面体的对称性引起的。作为这里开发的技术的应用，我们表明，Appell函数$F_4$不承认某种欧拉型整表示。
Monotonicity of ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions and associated Turán types inequalities
本文证明了$q$ -Kummer合流超几何函数和$q$ -超几何函数的某些比的单调性。这些结果也与Turán型不等式密切相关。为了获得主要结果，我们应用的方法开发的早期由作者的情况下，经典的库默尔和高斯超几何函数。
A New Class of Integrals Involving Extended Hypergeometric Function
本文研究了含有广义超几何函数的广义积分公式，所得结果用广义超几何函数表示.对于推广的高斯超几何函数和汇合型超几何函数，本文的主要结果也指出了某些特殊情况。
On solutions of the q-hypergeometric equation with q^{N}=1
考虑q^{N}=1的q-超几何方程和$α, β, γ\in {\Bbb Z}$ .我们在由幂级数乘以对数函数的幂给出的函数空间上求解这个方程。我们证明了在拟常数域上解的子空间是二维的。我们明确地得到了这个空间的基础。在此基础上，我们表示了Nishizawa和Ueno构造的Barnes型q-超几何函数。然后我们看到这个函数在原点有对数奇点。这是0< |q| <1和|q| =1的q-超几何函数之间的差异。
We describe a uniform way of obtaining basic hypergeometric functions as limits of the elliptic beta integral. This description gives rise to the construction of a polytope with a different basic hypergeometric function attached to each face of this polytope. We can subsequently obtain various relations, such as transformations and three-term relations, of these functions by considering geometrical properties of this polytope. The most general functions we describe in this way are sums of two very-well-poised ${}_{10}ϕ_9$'s and their Nassrallah-Rahman type integral representation.
In this paper, we obtain the analytical solutions of Laplace transforms based some novel integrals with suitable convergence conditions, by using hypergeometric approach (some algebraic properties of Pochhammer symbol and classical summation theorems of hypergeometric series ${}_{2}F_{1}(1)$, ${}_{2}F_{1}(-1)$ , ${}_{4}F_{3}(-1)$) . Also, we obtain the Laplace transforms of arbitrary powers of some finite series containing hyperbolic sine and cosine functions having different arguments, in terms of hypergeometric and Beta functions. Moreover, Laplace transforms of even and odd positive integra…
In this paper, we obtain recursion formulas for the Kampé de Feŕiet hypergeometric matrix function. We also give finite and infinite summation formulas for the Kampé de Feŕiet hypergeometric matrix function.
The aim of this note is to provide a new identity connected with the Gauss hypergeometric function. This is achieved using results of certain combinatorial identities and a hypergeometric function approach.
In this paper, we define a new type multivariable hypergeometric function. Then, we obtain some generating functions for these functions. Furthermore, we derive various families of multilinear and multilateral generating functions for these multivariable hypergeometric functions and their special cases are also given.
In this paper we derive the infinite summation formulas of Srivastava's general triple hypergeometric function. Certain particular cases leading to infinite summation formulas for fourteen Lauricella and three Srivastavaś triple hypergeometric functions are also presented.
The action of Hecke operators on hypergeometric functions
研究了Hecke算子在超几何函数集合上的作用。我们表明，这些运营商的频谱是一组的权力n^a和多项式发挥主导作用，在研究相应的本征函数。作为推论，我们得到了完全乘性超几何系数的一个刻画。
A Class of Extended Hypergeometric Functions and Its Applications
近年来，出现了不同形式的β函数和含额外参数的超几何函数。受这些思想的启发，本文首先引入广义超几何函数的一个新的扩张，然后给出一些基本结果。接下来，我们将导出某些函数的性质，如扩展的Gauss超几何函数，扩展的Appell的超几何函数$\mathbf{F}_{1}^{(κ_{l})}$，$\mathbf{F}_{2}^{(κ_{l})}$，和扩展的Lauricella的超几何函数$\mathbf{F}_{D,(κ_{l})}^{(r)}$，$\mathbf{F}_{A,(κ_{l})}^{(r)}$，包括变换公式，有限和表示，Mellin-Barnes型整表示和递归关系。利用本文提出的一些新的整表示，建立了一个涉及广义Gauss超几何函数的Hardy-Hilbert型不等式.
Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields.
In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.
Some integrals of hypergeometric functions
考虑一类涉及两个具有复杂自变量的经典超几何函数乘积的定积分。本文证明了这个积分不依赖于超几何函数的参数这一令人惊讶的事实。
Ramanujan's Master theorem for the hypergeometric Fourier transform on root systems
拉马努金主定理指出，在适当的条件下，交替幂级数的梅林变换为该级数的系数提供了一个插值公式。拉马努金应用这个定理计算了几个定积分和幂级数，这就解释了为什么它被称为“大师定理”。本文证明了根系上超几何傅里叶变换的Ramanujan Master定理的一个类似定理。这个定理推广到任意正重函数的结果以前证明的球面傅里叶变换半单黎曼对称空间。
An Extended k-type Hypergeometric Functions
超几何函数及其推广在各种应用中起着重要的作用。许多作者用多种方法建立了超几何函数的推广。本文旨在建立（可能是新的）扩展的$k-$型超几何函数$_{2}f_{1}^{k}[a, b; c; ω; z]$，并研究其各种性质，包括整表示、微分公式和分数次积分与微分公式.
Degenerate Gauss hypergeometric functions
这是一个终止和病态定义的高斯超几何函数的研究。相应的超几何方程有一组退化的24库默尔的解决方案。我们描述了这些解决方案和它们之间的关系。
Hypergeometric functions as generalized Stieltjes transforms
本文应用广义斯蒂尔切斯变换表示研究广义超几何函数。在这些结果中，我们证明了新的整表示、不等式、Padé表的性质以及广义超几何函数作为保角映射的性质.
Connection formulas for the confluent hypergeometric functions and the functional relation for the Hurwitz zeta function
利用汇合型超几何方程的联络问题证明了Hurwitz zeta函数的函数关系。
Properties of generalized univariate hypergeometric functions
基于Spiridonov对Gauss超几何函数的椭圆推广的分析，我们建立了一个7参数广义椭圆、双曲和三角一元超几何函数族的通用框架.在每一种情况下，我们使用适当的Nassrallah-Rahman β积分形式导出E_7型（椭圆、双曲）和E_6型（三角）外尔群下广义超几何函数的对称性，并使用θ函数和正弦函数的基本加法公式导出邻接关系。分别用Ruijsenaars的相对论性超几何函数和Askey-Wilson函数证明了双曲和三角超几何函数的顶层退化.我们表明，退化过程产生各种新的和已知的身份双曲和三角特殊函数。我们还描述了一个密切的双曲和三角理论之间的联系，这产生了一个双曲超几何函数的表达作为一个明确的双线性和三角超几何函数。
Representation of the Gauss hypergeometric function by multiple polylogarithms and relations of multiple zeta values
本文描述了高斯超几何方程$F(α,β,γ;z)$的一种解，其系数为多个多项式的$\Z$线性组合的参数$α,β,γ$的幂级数。并利用超几何方程解的表示和联络关系，给出了多重zeta值的一些关系。
We express explicitly the Heckman-Opdam hypergeometric function for the root system of type A with a certain degenerate parameter in terms of the Lauricella hypergeometric function.
We examine the power-series solutions and the series solutions in terms of the Hermite functions for the biconfluent Heun equation. Infinitely many cases for which a solution of the biconfluent equation is presented as an irreducible linear combination of four generalized hypergeometric functions, that in general do not reduce to polynomials, are identified.
Numerical Evaluation of Generalized Hypergeometric Functions for Degenerated Values of Parameters
本文给出了参数退化值的广义超几何函数${}_p F_{p-1}$的联络公式的一个生成算法。我们还表明，这些连接公式给出了一个快速的方法数值计算的广义超几何函数附近的$\infty$。
Motivated by the work on hypergeometric summation theorems (recorded in the table III of Prudnikov et al. pp. 541-546), we have established some new summation theorems for Clausen's hypergeometric functions with unit argument in terms of $π$ and natural logarithms of some rational and irrational numbers. Also, we have given some modified summation theorems for Clausen's hypergeometric functions.
We consider the q-derivatives of the Srivastava and Daoust basic multivariable hypergeometric function with respect to the parameters. This function embodies a entire number of various q-hypergeometric series of one and several variables. Explicit equations are given for general case of summation indexes with positive real coefficients. As an example derivatives of q-analog of non-confluent Horn type hypergeometric function $H_3$ is presented.
Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients were constructed recently. These fundamental solutions are directly connected with multiple Lauricella hypergeometric function and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, some relations following from the decomposition formula for one multidimensional Lauricell hypergeometric function are determined.
From the algebraic solution of $x^{n}-x+t=0$ for $n=2,3,4$ and the corresponding solution in terms of hypergeometric functions, we obtain a set of reduction formulas for hypergeometric functions. By differentiation and integration of these results, and applying other known reduction formulas of hypergeometric functions, we derive new reduction formulas of special functions as well as the calculation of some infinite integrals in terms of elementary functions.
On an extension of extended Beta and hypergeometric functions
主要是出于某些有趣的最近的Gamma，Beta和超几何函数的扩展，我们在这里介绍新的β函数，超几何和汇合型超几何函数的扩展。我们系统地研究了这些扩展函数的几个性质，即它们的各种整表示，梅林变换，导数，变换，求和公式和渐近性。文中还指出了本文主要结果的某些特殊情况的相关联系。
A formula for the nonsymmetric Opdam's hypergeometric function of type $A_2$
本文给出了$A_2$型非对称Heckman-Opdam超几何函数的一个显式公式。这是通过微分相应的对称超几何函数得到的。
On some linear combination of two contiguous hypergeometric functions
本文的目的是利用某些邻接关系，给出两个对称邻接超几何函数的线性组合在参数一定条件下的渐近性态。
Representations and inequalities for generalized hypergeometric functions
通过广义Stieltjes变换、Laplace变换和余弦傅里叶变换，统一了已知的表示，找到了广义超几何函数的整表示.利用正性条件的重量在这个表示，我们建立各种新的事实，广义超几何函数，包括完全单调性，对数凸性的上参数，单调性的比率和新的证明卢克的界限。此外，我们还利用Bessel型超几何函数的级数表示，得到了它们的双边不等式。
In this paper, we study some extended hypergeometric functions from matrix point of view. We have given the integral representations of these matrix functions. Finally, we obtain some generating function relations using fractional derivative operators.
Parametric transformations between the Heun and Gauss hypergeometric functions
超几何函数和Heun函数是经典的特殊函数。它们之间的变换公式通常由它们的微分方程关于某些覆盖P1-to-P1的拉回变换导出。这给出了Heun函数的表达式，用更好理解的超几何函数表示。本文给出了具有自由连续参数的超几何到Heun拉回变换的列表，并通过Heun到超几何的约化公式说明了其中的大多数。总共有61个参数变换，最大度为12。
Transformations and invariants for dihedral Gauss hypergeometric functions
具有二面角单值群的超几何方程可以用初等函数求解。本文利用Clausen公式的推广和双超几何和的终止，给出了这类超几何方程的二次单值不变量的显式通解。此外，还给出了二面体超几何方程的拉回变换，包括有限（二面体）单值群方程的Klein拉回变换。
Hypergeometric functions and the Tricomi operator
我们展示了某些超几何函数如何在寻找广义Tricomi算子的基本解中发挥重要作用。
Inspired by certain interesting recent extensions of the gamma, beta and hypergeometric matrix functions, we introduce here new extension of the gamma and beta matrix function. We also introduce new extensions of the Gauss hypergeometric matrix function, confluent hypergeometric matrix function, Appell matrix function and Lauricella matrix function of three variables in terms of the new extended beta matrix function. Then we investigate certain properties of these extended matrix functions such as the integral representations, differential formulae and recurrence relations.
Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves
本文给出了$_2F_1$ -超几何函数的新的代数变换。在我们的方法中的主要新颖之处是$_2F_1$ -超几何函数之间的身份不同志村曲线之间的自守形式的身份的解释。
A Connection Formula for the $q$-Confluent Hypergeometric Function
给出了$q$ -汇合型超几何函数${}_2φ_1(a,b;0;q,x)$的一个联络公式.将我们的联络公式与张氏关于${}_2φ_0(a,b;-;q,x)$的联络公式相结合，得到了矩阵形式的$q$ -汇合型超几何方程的联络公式.并通过取我们的连接公式的极限$q\to 1^{-}$，得到了Kummer汇合型超几何函数的连接公式。
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