A. M. Ishkhanyan
We present infinitely many solutions of the general Heun equation in terms of generalized hypergeometric functions. Each solution assumes that two restrictions are imposed on the involved parameters: a characteristic exponent of one of the singularities should be a non-zero integer, and the accessory parameter must satisfy a polynomial equation.
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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。
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.
A classification of coverings yielding Heun-to-hypergeometric reductions
Heun和Gauss超几何方程之间的拉回变换给出了Heun函数在更好理解的超几何函数方面的有用表达式。本文分类，莫比乌斯自同构，覆盖P1到P1，产生拉回变换从超几何Heun方程至少有一个自由参数（不包括情况下，涉及的超几何方程有循环或二面角单值）。在所有，61参数hypergeometric-to-Heun变换被发现，最大程度的12。其中，28个拉回是超几何函数和Heun函数之间的较小次数变换的组合。61个变换由48个不同的Belyi覆盖实现（尽管2个覆盖应该计数两次，因为它们的模域是二次的）。同样的Belyi覆盖物出现在其他几个环境中。例如，38个覆盖出现在Herfutner的列表中的椭圆曲面P1与四个奇异纤维，作为其j-不变量。顺便说一下，我们展示了一种优雅的方法来证明不存在具有某些分支模式的覆盖P1-to-P1。
We show that there exist infinitely many particular choices of parameters for which the three-term recurrence relations governing the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions become two-term. In these cases the coefficients are explicitly written in terms of the gamma functions.
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 show that there exist infinitely many nontrivial choices of parameters of the single confluent Heun equation for which the three-term recurrence relations governing the expansions of the solutions in terms of the confluent hypergeometric functions 1F1 and 0F1 are reduced to two-term ones. In such cases the expansion coefficients are explicitly calculated in terms of the Euler gamma functions.
Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions
研究了合流Heun方程的解关于三组不同的Kummer汇合型超几何函数的级数展开。展开式的系数一般服从定义双边无穷级数的三项递推关系;然而，在特定情况下，四项和两项关系也是可能的。讨论了导出列的左侧和/或右侧终止的条件。
Degenerate and dihedral Heun functions with parameters
正如高斯超几何函数一样，局部Heun函数的特殊情况可以是Liouvillian（即“初等”）函数。获得这些函数的一种方法是通过具有Liouvillian解的高斯超几何方程的拉回变换。本文给出了Heun方程的Liouvillian解，Heun方程是具有循环或二面体单值群的参数超几何方程的拉回。
Several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions are presented. Three different sets of these functions are examined. Discussing the expansions without a pre-factor, it is shown that two of these functions provide expansions the coefficients of which obey three-term recurrence relations, while for the third confluent hypergeometric function the corresponding recurrence relation generally involves five-terms. The latter relation is reduced to a three-term one only in the case when the double-confluent Heun equati…
New transformation law for Heun and Hypergeometric Equations
本文建立了Heun-to-Heun变换和Heun-to-Hypergeometric变换的新形式。通过以非线性方式改变自变量来实现变换。使用这些，我们还指出了一些简单的例子，方程之间的转换，不是Fuchsian和概括Heun超几何变换。
We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions …
On reducing the Heun equation to the hypergeometric equation
列举了Heun方程通过自变量的多项式变换化为超几何方程的各种方法，并对它们进行了分类。Heun到超几何的约化类似于经典的超几何恒等式，但约化存在的条件涉及超几何方程所不具备的Heun方程的特征，即它的交比和辅助参数。减少包括二次和三次变换，这可能是执行只有当奇点的Heun方程形成一个谐波或等谐波四倍，分别;和几个更高的程度的转换。这一结果修正和推广了前一篇文章中只发现了二次变换的一个定理。[See K. Kuiken，“Heun's equation and the hypergeometric equation”，SIAM Journal on Mathematical Analysis 10：3（1979），655-657.
Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial
我们研究Heun的微分方程的情况下，其中一个奇点是明显的。特别地，我们推测了与广义超几何微分方程的关系，并在某些情况下建立了它。我们应用我们的结果例外的雅可比多项式。
We consider some hypergeometric functions and prove that they are elementary functions. Consequently, the second order moments of Meyer-Konig and Zeller type operators are elementary functions. The higher order moments of these operators are expressed in terms of elementary functions and polylogarithms. Other applications are concerned with the expansion of certain Heun functions in series or finite sums of elementary hypergeometric functions.
Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions
利用第一类二元不不完全β函数和Appell广义超几何函数构造了合流Heun方程解的几种展开式.不同展开式的系数服从四项、五项或六项递归关系，这些递归关系仅在少数例外情况下被简化为涉及较少项的关系。讨论了通过级数的终止获得有限和解的条件。
The Heun's equation is the Fuchsian equation of second order with four regular singularities. Heun functions generalize well-known special functions such as Spheroidal Wave, Lamé, Mathieu, hypergeometric-type functions, etc. The recursive relation of coefficients starts to appear by putting a power series into the Heun equation. A local Heun function consists of the three term recurrence relation in its power series, and we prove that the function is not convergent on the boundary of the disc of convergence.
Special Solutions of $q$-Heun Equation by $q$-Integral Transformations
通过考虑多项式型解的 $q$-积分变换，得到了 $q$-Heun 方程表示为 $q$-超几何函数有限求和的特殊解。
We present a conspicuous number of indefinite integrals involving Heun functions and their products obtained by means of the Lagrangian formulation of a general homogeneous linear ordinary differential equation. As a by-product we also derive new indefinite integrals involving the Gauss hypergeometric function and products of hypergeometric functions with elliptic functions of the first kind. All integrals we obtained cannot be computed using Maple and Mathematica.
P-symbols, Heun Identities, and 3F2 Identities
探讨了Riemann P-符号在推导包含参数化特殊函数H1的恒等式中的有用性。H1是Heun方程的解析局部解，Heun方程是黎曼球面上具有四个正则奇点的正则二阶微分方程。所讨论的恒等式包括来自Moebius自同构和F-同伦的恒等式，以及二次和双二次变换的恒等式。研究了H1与3F 2型广义超几何函数相同的情形，导出了3F 2（a 1，a 2，e+1; b 1，e;x）的Pfaff变换和欧拉变换.它们推广了Bailey和Slater的几个3F 2恒等式。
The 192 Solutions of the Heun Equation
给出了Heun方程的192个局部解的机器生成列表。它们类似于库默尔的高斯超几何方程的24个解，因为这两个方程分别是黎曼球面上具有四个和三个奇点的正则Fuchsian微分方程。通过将具有n个奇点的方程的自同构群确定为Coxeter群D_n，使制表变得容易。192个表达式中的每一个都由D_4的元素标记。其中24个是局部Heun函数H1的等价表达式，并证明了H1的24阶变换群同构于对称群S_4.同构将每个变换编码为抽象四元素集的置换，而不是与奇点集相同。
A series solution of the general Heun equation in terms of incomplete Beta functions
我们证明了当无穷远奇点的特征指数为零时，一般Heun方程的解可以用不完全β函数展开.通过级数的终止，得到了两组无穷参数的封闭解。这些有限和的解决方案是书面的初等函数，一般是准多项式。展开式的系数服从三项递推关系，在某些特殊情况下可能变为两项。我们讨论的情况下，当递归关系涉及到两个非连续的条款，并表明，系数的扩展，然后明确计算和一般的解决方案的Heun方程的构造作为一个组合的几个超几何函数与准多项式前因子。
The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a ``time variable'' $t$. From ladder operators one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when $n$, the order of the Hankel matrix and also t…
Asymptotic behavior of Heun function and its integral formalism
The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric functions. Heun functions are applicable to diverse areas such as theory of black holes , lattice systems in statistical mechanics , solution of the Schrodinger equation of quantum mechanics , and addition of three quantum spins. In this paper, applying three term recurrence formula, I consider asymptotic behaviors of Heun function and its integral formalism including all higher terms of A_n's. I will show how the power series expansion of Heun functions can be converted to…
Potentials of the Heun class: the triconfluent case
自从量子力学出现以来，人们成功地开发了不同的方法来寻找薛定谔方程的解析解。在这里，我们遵循并推广了Natanzon和其他人开创的方法，通过该方法，薛定谔方程可以转换为另一个众所周知的超越函数方程（例如，超几何方程）。这就设置了一个可以进行这种变换的势类。我们的推广包括在寻找潜在的允许转换的薛定谔方程到一个triconfluent Heun方程。我们发现这类潜在的能量本征值，本征函数和确切的超伙伴。
Apparent singular points of factors of reducible generalized hypergeometric equations
考虑一个可约的广义超几何方程，其子方程具有明显的奇点。我们确定多项式的根是这些点。我们证明了这个多项式是一个广义超几何多项式。关键词和短语。广义超几何方程，可约，视奇点，朗斯基函数的子式。
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.
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$\infty$。
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.
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。
On the application of a hypergeometric identity to generate generalized hypergeometric reduction formulas
本文通过新的超几何恒等式推导出新的求和公式，并验证了相关函数的递推关系
The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations
通过推广Darboux变换，得到Heun方程的等单值变换，并找到了具有相同单值结构的Heun方程对和三重方程。通过构造广义达布变换，我们建立了一个新的可换算子的构造，它保证了有限间隙性质。作为应用，我们在第三部分证明了这些定理。
We solve connection problem between fundamental solutions at singular points $0$ and $1$ for the generalized hypergeometric function, using analytic continuation of the integral representation. All connection coefficients are products of the sine and the cosecant.
The paper studies logarithmic convexity and concavity of the generalized hypergeometric function with respect to simultaneous shift of several parameters. We use integral representations and properties of Meijer's $G$ function to prove log-convexity. When all parameters are shifted we use series manipulations to examine the power series coefficients of the generalized Turánian formed by the generalized hypergeometric function. In cases when all zeros of the generalized hypergeometric function are real, we further explore the consequences of the extended Laguerre inequalities and formulate a co…
As for the positivity of ${}_1F_2$ generalized hypergeometric functions, we present a list of necessary and sufficient conditions in terms of parameters and determine the region of positivity by certain Newton diagram.
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。
Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions
1993年，作者之一给出了指数级数截面比单调性的一些证明。它们导致了关于Kummer超几何函数的比单调性的更一般的猜想，但自1993年以来一直没有得到证明。本文在更一般的情形下证明了Kummer超几何函数的这些性质及其对Gauss和广义超几何函数的进一步推广。这些结果也与Turan型不等式密切相关。
Galois groups of the Lie-irreducible generalized $q$-hypergeometric equations of order three with $q$-real parameters : an approach using a density theorem
本文利用Sauloy的一个密度定理计算了具有q-实参数的三阶Lie-不可约正则奇异广义q-超几何方程的差分伽罗瓦群.与微分情形相比，我们证明了这些群自动包含特殊线性群SL（3，C）。
Voros coefficients of the generalized hypergeometric differential equations with a large parameter are defined and their explicit forms are given for the origin and for the infinity. It is shown that they are Borel summable in some specified regions in the space of parameters and their Borel sums in the regions are given.
We study a connection problem between the fundamental systems of solutions at singular points $0$ and $1$ for the generalized hypergeometric equation which is satisfied by the generalized hypergeometric series ${}_nF_{n-1}$. In general, the local solution space around $x=1$ consists of one dimensional singular solution space and $n-1$ dimensional holomorphic solution space. Therefore in the case of $n\ge3$, the expression of connection matrix depends on the choice of the fundamental system of solutions at $x=1$. On the connection problem for ordinary differential equations, Schäfke and Schmidt…
A few identities and integrals involving Pochhammer symbols, Jacobi polynomials, and the generalized hypergeometric function
本文通过Jacobi多项式和广义超几何函数的展开，推导出涉及Pochhammer符号的新的恒等式和积分闭形式。
Biconfluent Heun equation (BHE) is a confluent case of the general Heun equation which has one more regular singular points than the Gauss hypergeometric equation on the Riemann sphere $\hat{\mathbb{C}}$. Motivated by a Nevanlinna theory (complex oscillation theory) approach, we have established a theory of \textit{periodic} BHE (PBHE) in parallel with the Lamé equation verses the Heun equation, and the Mathieu equation verses the confluent Heun equation. We have established condition that lead to explicit construction of eigen-solutions of PBHE, and their single and double orthogonality, and …
Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators
对于高斯超几何函数的八个$n$阶导数参数变换公式，分别给出了相应的分数阶积分公式。对于这两种类型的公式，微分或积分算子都交织在超几何微分算子的两个作用之间（对于两组参数）：所谓的嬗变性质。这导致八个分数积分公式和四个广义斯蒂尔切斯变换公式的超几何微分方程的六个不同的显式解的每一个，通过让变换的解决方案。通过特殊化得到了六个解的两个Euler型整表示。
Reduction formulas for sums of products of hypergeometric functions can be traced back to Euler. This topic has an intimate connection to summation and transformation formulas, contiguous relations and algebraic properties of the (generalized) hypergeometric differential equation. Over recent several years, important discoveries have been made in this subject by Gorelov, Ebisu, Beukers and Jouhet and Feng, Kuznetsov and Yang. In this paper, we give a generalization of Feng, Kuznetsov and Yang identity covering also Ebisu's and Gorelov's formulas as particular cases.
We derive a convergent expansion of the generalized hypergeometric function ${}_{p-1}F_p$ in terms of the Bessel functions ${}_{0}F_1$ that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We further obtain a convergent expansion of the generalized hypergeometric function ${}_{p}F_p$ in terms of the confluent hypergeometric functions ${}_{1}F_1$ that holds uniformly in any right half-plane. For both functions, we make a further step and give convergent expansions in terms of trigonometric, exponential and rational functions that hold uniformly in the s…
The analytic solution for the power series expansion of Heun function
Heun函数推广了所有著名的特殊函数，如球面波、Lame、Mathieu和超几何_2F_1、_1F_1和_0F_1函数。Heun函数适用于不同的领域，如黑洞理论，统计力学中的晶格系统，量子力学中薛定谔方程的解，以及三个量子自旋的添加。本文将应用三项递推公式（Choun，Y.S.，arXiv：1303.0806.，2013）推广到Heun函数（无穷级数和多项式）的封闭形式的幂级数展开，其中包含A_n的所有高阶项。第三部分是对Heun函数幂级数展开式在一篇论文中的应用的分析。（R.S. Maier，Math.Comp.33，2007）由于篇幅限制，192个Heun函数的最终方程未包含在本文中，但请随时与我联系以获得最终解决方案。第四节给出了Heun函数幂级数展开式的两个应用实例。本文是“特殊函数与三项递推公式（3TRF）”系列文章10篇中的第3篇。请参阅第5节中的所有论文系列。本文讨论了三项递推公式（3TRF）。下一篇文章在该系列描述积分形式的Heun函数及其渐近行为的分析。
Our purpose in this present paper is to investigate generalized integration formulas containing the extended generalized hypergeometric function and obtained results are expressed in terms of extended hypergeometric function. Certain special cases of the main results presented here are also pointed out for the extended Gauss' hypergeometric and confluent hypergeometric functions.
We prove a duality relation for generalized basic hypergeometric functions. It forms a $q$-extension of a recent result of the second and the third named authors and generalizes both a $q$-hypergeometric identity due to the third named author (jointly with Feng and Yang) and a recent identity for the Heine's ${}_2ϕ_{1}$ function due to Suzuki. We further explore various consequences of our identity leading to several presumably new multi-term relations for both terminating and non-terminating generalized basic hypergeometric series. Moreover, we give confluent versions of our results and furni…
The indefinite integral $$ \int x^αe^{ηx^β}\,_pF_q (a_1, a_2, \cdot\cdot\cdot a_p; b_1, b_2, \cdot\cdot\cdot, b_q; λx^γ)dx, $$ where $α, η, β, λ, γ\ne0$ are real or complex constants and $_pF_q$ is the generalized hypergeometric function, is evaluated in terms of an infinite series involving the generalized hypergeometric function. Related integrals in which the exponential function $e^{ηx^β}$ is either replaced by the hyperbolic function $\cosh\left(ηx^β\right)$ or $\sinh\left(ηx^β\right)$, or the sinusoidal function $\cos\left(ηx^β\right)$ or $\sin\left(ηx^β\right)$, are also evaluated in te…
The generalized Zwegers' $μ$-function and transformation formulas for the bilateral basic hypergeometric series
本文利用Slater和Bailey的双边基本超几何级数变换公式，推导出广义Zwegers' $μ$-函数的平移公式与新表达，并以此获得了Weierstrass和Jacobi椭圆函数的新Fourier展开。
Inequalities and monotonicity of ratios for generalized hypergeometric function
本文证明了具有正参数的${_{q+1}}F_{q}(-x)$形式的广义超几何函数在某些附加条件下的双边不等式。下限和上限均与正半轴端点处的${_{q+1}}F_{q}(-x)$值一致，并且在其中一个端点处渐近精确。这些不等式是由证明两个带位移参数的广义超几何函数的商的单调性的定理导出的。证明的关键是广义超几何函数的广义Stieltjes表示。该表示还提供了另一种方法来推导${_{3}F_{2}}(1)$的第二类关系，并导致根据Appell函数$F_3$的${_{4}F_{3}}(x)$的整表示。在论文的最后一部分，我们列出了一些开放的问题和建议。
The main goal of this paper is to derive a number of identities for the generalized hypergeometric function evaluated at unity and for certain terminating multivariate hypergeometric functions from the symmetries and other properties of Meijer's $G$ function. For instance, we recover two- and three-term Thomae relations for ${}_3F_2$, give two- and three-term transformations for ${}_4F_3$ with one unit shift and ${}_5F_4$ with two unit shifts in the parameters, establish multi-term identities for general ${}_{p}F_{p-1}$ and several transformations for terminating Kampé de Fériet and Srivastava…
Analytic continuation of the generalized hypergeometric series near unit argument with emphasis on the zero-balanced series
综述了获得超几何级数$_{p+1}F_p(z)$在$z=1$附近的解析延拓的各种方法，并给出了一些结果.一种方法是建立一个关于$p$的递归关系，然后，在它的重复使用后，求助于高斯超几何级数的众所周知的性质。另一种方法是基于基本的广义超几何微分方程的性质：对于$z=1$周围的幂级数展开中的系数，一个对任何$p$都有效的通用公式，被发现在单位辐角的终止萨尔舒茨超几何级数的极限方面。然后，在应用萨尔许茨超几何级数的适当变换公式之后，可以获得每个特定$p$的最终结果。零平衡超几何级数在单位辐角的行为，近年来受到特别的关注，进行了更详细的讨论。有关的问题，涉及的行为，部分和这样的系列简要说明。
In this present paper our aim is to deal with two integral transforms which involving the Gauss hypergeometric function as its kernels. We prove some compositions formulas for such a generalized fractional integrals with k Bessel function. The results are established in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some corresponding assertions for Riemann Liouville and Erdelyi Kober fractional integral transforms.
The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer's $G$-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer's $G$-fun…
Two closed-form evaluations for the generalized hypergeometric function ${}_4F_3(\frac1{16})$
本研究通过拆分$_3F_2$的奇偶分量并结合已知$_3F_2(\pm\frac14)$结论，得到自变量为$1/16$的$_4F_3$的两种闭式求值，还推出$\pi^2$的无穷和表示并探讨了相关研究关联。
By building a second order adjoint difference equations on nonuniform lattices, the generalized Rodrigues type representation for the second kind solution of a second order difference equation of hypergeometric type on nonuniform lattices is given. The general solution of the equation in the form of a combination of a standard Rodrigues formula and a generalized Rodrigues formula is also established.
The authors establish the necessary and sufficient conditions under which certain combinations of Gaussian hypergeometric function and elementary function are monotone in the parameter, which generalize the recent results of generalized elliptic integrals of the first and second kinds obtained by Qiu et al. Moreover, the authors also prove two monotonicity theorems of generalized elliptic integrals from another point of view.
In this paper, we aim to present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella hypergeometric functions, by using the extended Bessel function due to Boudjelkha [4]. Some recurrence relations, transformation formulas, Mellin transform and integral representations are obtained for these generalizations. Further, an extension of the Riemann-Liouville fractional derivative operator is established.
Partial sums of hypergeometric series of unit argument
研究单位宗量广义超几何级数部分和的渐近性态。
On digamma series convertible into hypergeometric series
本文通过适当的极限过程，将单位值处的广义超几何函数多项恒等式转化为与双伽马级数相连的恒等式，从而为该函数关于所有参数的梯度的$1$-范数提供了文献中尚无类比的超几何表达式。
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