Mikhail Yu. Kalmykov, Bernd A. Kniehl
We briefly sketch a proof concerning the structure of the all-order epsilon-expansions of generalized hypergeometric functions with special sets of parameters.
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Series and epsilon-expansion of the hypergeometric functions
讨论了与一元超几何函数的ε展开有关的多重和的解析计算的最新进展。
Higher-order diagrams required for radiative corrections to mixed electroweak and QCD processes at the LHC and anticipated future colliders will require numerically stable representations of the associated Feynman diagrams. The hypergeometric representation supplies an analytic framework that is useful for deriving such stable representations. We discuss the reduction of Feynman diagrams to master integrals, and compare integration-by-parts methods to differential reduction of hypergeometric functions. We describe the problem of constructing higher-order terms in the epsilon expansion, and cha…
Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters
证明了如下定理：1）Gauss超几何函数2F_1的Laurent展开式（I_1 + a * b，I_2 + b * c; I_3 + p/q + c * c; z），2F1（I_1 + p/q + a * b; I_2 + p/q + b * c; z），2F1（I_1 + p/q + a~*，I_2 + b~*，I_3 + p/q + c~*，I_3 + p/q + c~*）; z），其中I_1，I_2，I_3，p，q为任意整数，a，b，c为任意数，λ为无穷小参数，可表示为系数为多项式比的q-单位根的多重多项式; 2）高斯超几何函数2F1的Laurent展开（I_1 + p/q + a * k，I_2 + b * k; I_3 + c * k; z）可表示为系数为多项式之比的q-单位根乘对数幂的多重多项式; (2))和多个有理数和（见Eq. (3))可表示在多个多项式; 4）广义超几何函数（见方程。(4))可以用多个多项式来表示，其系数是多项式的比率。
Differential Reduction Algorithms for the All-Order Epsilon Expansion of Hypergeometric Functions
超几何函数提供了费曼图的一个有用的表示出现在精密唯象学。在维数正则化中，需要这些函数关于d=4的ε展开。我们讨论了差分约简算法的现状。作为一个例子，我们考虑了Appell超几何函数关于参数的整数值的全阶ε-展开的构造，并给出了前几项的显式估计.
On the all-order epsilon-expansion of generalized hypergeometric functions with integer values of parameters
我们继续研究超几何函数的ε-展开式的解析系数的构造及其与费曼图的联系。本文应用求超几何函数微分方程迭代解的方法，证明了如下结果（定理1）：参数为整数值的广义超几何函数的ε展开式可用系数为多项式比的广义多项式表示。本文的证明方法为计算Laurent展开式的高阶系数提供了一种有效的算法。
Assumed that the parameters of a generalized hypergeometric function depend linearly on a small variable $\varepsilon$, the successive derivatives of the function with respect to that small variable are evaluated at $\varepsilon=0$ to obtain the coefficients of the $\varepsilon$-expansion of the function. The procedure, quite naive, benefits from simple explicit expressions of the derivatives, to any order, of the Pochhammer and reciprocal Pochhammer symbols with respect to their argument. The algorithm may be used algebraically, irrespective of the values of the parameters. It reproduces the …
It is demonstrated that the well-regularized hypergeometric functions can be evaluated directly and numerically. The package NumExp is presented for expanding hypergeometric functions and/or other transcendental functions in a small regularization parameter. The hypergeometric function is expressed as a Laurent series in the regularization parameter and the coefficients are evaluated numerically by using the multi-precision finite difference method. This elaborate expansion method works for a wide variety of hypergeometric functions, which are needed in the context of dimensional regularizatio…
All order epsilon-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters
证明了下列高斯超几何函数2F 1的Laurent展开式（I1+a* p，I2+b*ep; I3+c* p;z），2F1（I1+a* b，I2+b* c;I3+1/2+c* c;z），2F1（I1+1/2+a* b，I2+b* c; I3+c* c;z），2F1（I1+1/2+a* b; I2+b* c; I3+1/2+c* c;z），2F1（I1+1/2+a*，I2+1/2+b*; I3+1/2+c*; z），其中I1、I2、I3为任意非负数整数，a、b、c为任意数，且λ为任意小参数，的调和多项式的Remiddi和Vermaseren的多项式系数。构造了一种计算Laurent展开式高阶系数的有效算法。本文还讨论了高斯超几何函数的一些特殊情况。
In this talk, we discuss the algorithm for the construction of analytical coefficients of higher order epsilon expansion of some Horn type hypergeometric functions of two variables around rational values of parameters.
Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
我们继续研究超几何函数的ε-展开式的解析系数的构造及其与费曼图的联系。在本文中，我们证明了以下结果：定理A：多个（逆）二项式和的任意重量和深度（见方程。（1.1））可以用Remiddi-Vermaseren函数表示。定理B：具有一个半整数参数值的超几何函数的展开式（见等式2）。（1.2））可以用Remiddi和Vermaseren的调和多项式来表示，其系数是多项式的比率。一些额外的材料可以通过http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html获得。
Hypergeometric functions, their epsilon expansions and Feynman diagrams
我们回顾了费曼图的超几何函数方法。特别考虑到建设的洛朗扩展。作为一个例子，我们描述了一个收集的物理上重要的单圈顶点图，这种方法是有用的。
Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
将参数为任意值的高斯超几何函数2F 1约化为两个参数为固定值的函数，这两个函数与原函数相差整数。证明了在参数为整数和/或半整数值的情况下，只有三种代数独立的高斯超几何函数。这种类型（在我们的分类中为F型）的函数的ε展开要求引入与椭圆函数的推广相关的新函数。对于其他五种类型的功能的高阶ε展开的功能的权重4的构建。展开的结果只能用Nielsen多项式表示。考虑了单重线q无质量线q圈离壳传播子图和双重线q-1无质量线q圈泡的约化和ε展开。该代码（Mathematica/FORM）可通过网址http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html获得
HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: the first one, FdFunction, for manipulations with Appell hypergeometric functions $F_D$ of $r$ variables; and the second one, FsFunction, for manipulations with Lauricella-Saran hypergeometric functions $F_S$ of three variables. Both functions are related with one-loop Feynman diagrams. The published version includes also Chapter 5 with two theorems about structure of coefficients of epsilon-expansion of the Ho…
We briefly discuss the transcendental constants generated through the epsilon-expansion of generalized hypergeometric functions and their interrelation with the "sixth root of unity."
Geometrical approach to Feynman integrals and the epsilon-expansion
应用几何启发表示的ε展开的两点函数与不同的质量被认为是。对于展开式中的任意项，用对数正弦积分给出了显式结果。本文还讨论了三点情形下ε展开式的构造。
The decomposition in partial fractions of the quotient of Pochhammer symbols improves considerably a method, suggested in a precedent paper, which allows one to obtain the $\varepsilon$-expansion of functions of the hypergeometric class. The procedure is applied to several Appell and Kampé de Fériet functions considered in the literature. Explicit expressions and interesting properties of the derivatives of the Pochhammer and reciprocal Pochhammer symbols, which are essential elements in the procedure, are given in an appendix.
About higher order epsilon-expansion of some massive two- and three-loop master-integrals
对于某些维数调节的大量的二圈和三圈传播子型图的高阶ε展开构造。
We present a new methodology, suitable for implementation on computer, to perform the $ε$-expansion of hypergeometric functions with linear $ε$ dependent Pochhammer parameters in any number of variables. Our approach allows one to perform Taylor as well as Laurent series expansion of multivariable hypergeometric functions. Each of the coefficients of $ε$ in the series expansion is expressed as a linear combination of multivariable hypergeometric functions with the same domain of convergence as that of the original hypergeometric function. We present illustrative examples of hypergeometric func…
Some remarks on the epsilon-expansion of dimensionally regulated Feynman diagrams
本文讨论了维数调节的费曼积分的ε展开式的构造问题。对于某些类型的图，ε展开式的任意项可以用与多项式相关的对数正弦积分来表示。它示出了如何解析这些功能的延续，可以构造在广义尼尔森多项式。
We evaluate three typical four-loop non-planar massless propagator diagrams in a Taylor expansion in dimensional regularization parameter $ε=(4-d)/2$ up to transcendentality weight twelve, using a recently developed method of one of the present coauthors (R.L.). We observe only multiple zeta values in our results.
Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
本文利用微分方程方法和规范基约化，解决了指标依赖于小参数的超几何函数的Laurent展开问题，特别处理了涉及非平凡变量代换的有理指标情形，并提供了Mathematica包Diogenes实现算法。
The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the construction of the $ε$-expansion. As an example, we present a detailed discussion of the construction of the epsilon-expansion of the Appell function $F_3$ around rational values of parameters via an iterative solution of differential equations. As a by-product, we have found that the one-loop massless pentagon diagram in dimension $d=3-2ε$ is not expressible in terms of multiple polylogarithms. Another interesting example is the Puiseux-type solutio…
Starting from the equation obeyed by the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the Appell generalized hypergeometric functions of two variables of the fist kind. Several cases when the expansions reduce to ones written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; howeve…
We introduce new hypergeometric series expansions of the solutions to the general Heun equation. The form of the Gauss hypergeometric functions used as expansion function differs from that used before. We derive three such expansions and further generate, by termination of the series, closed-form solutions for several sets of involved parameters.
$\texttt{MultiHypExp}$: A Mathematica Package For Expanding Multivariate Hypergeometric Functions In Terms Of Multiple Polylogarithms
本文提出 Mathematica 程序包 MultiHypExp，通过微分算子与 Pfaffian 系统方法将多元超几何函数在维数正规化参数中展开为多重多对数，以处理多圈多尺度 Feynman 积分。
We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the fist kind. The coefficients of different expansions obey four-, five-, or six-term recurrence relations that are reduced to ones involving less number of terms only in a few exceptional cases. The conditions for deriving finite-sum solutions via termination of the series are discussed.
Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {\tt Si…
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。
We apply exact WKB methods to the study of the partition function of pure N=2 epsilon_i-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in epsilon_2/epsilon_1 (i.e. at large central charge) and in an expansion in epsilon_1. We find corrections of the form ~ exp[- SW periods / epsilon_1] to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of epsilon_1 when passing from the Nekrasov form of the partition function to the topological string t…
The canonical partition function of a two-dimensional lattice gas in a field of randomly placed traps, like many other problems in physics, evaluates to the Gauss hypergeometric function ${}_2F_1(a,b;c;z)$ in the limit when one or more of its parameters become large. This limit is difficult to compute from first principles, and finding the asymptotic expansions of the hypergeometric function is therefore an important task. While some possible cases of the asymptotic expansions of ${}_2F_1(a,b;c;z)$ have been provided in the literature, they are all limited by a narrow domain of validity, eithe…
We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental solution to the discrete Laplacian. For the proofs we express the resolvent in a general dimension in terms of the Appell--Lauricella hypergeometric function of type $C$ outside a disk encircling the spectrum. In low dimensions it reduces to a generalized hypergeometric function, for which certain transformation formulas are available for the desired expansions.
We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations depending on the same variables and with the same region of convergence as for original Horn function. The derivatives of Appell functions, generalized hypergeometric functions, confluent and non-confluent Lauricella series and generalized Lauricella series are explicitly presented. Applications to the calculation of Feynman diagrams are discussed, especial…
Partial wave expansion of the Coulomb-distorted plane wave is determined and studied. Dominant and sub-dominant asymptotic expansion terms are given and leading order three-dimensional asymptotic form is derived. The generalized hypergeometric function $_2F_2(a,a;a+l+1,a-l;z)$ is expressed with the help of confluent hypergeometric functions and the asymptotic expansion of $_2F_2(a,a;a+l+1,a-l;z)$ is simplified.
Hypergeometric-like Representation of the Zeta-Function of Riemann
我们提出了一个新的扩展的ζ函数的黎曼。目前的形式主义-它结合了插值与约束的思想和超几何函数的概念-可以以一种自然的方式，在霍金的ζ函数理论中推广，从而在量子场论，量子宇宙学和统计力学中提供各种应用。
On Powers of a Hypergeometric Function
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.
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.
On hypergeometric series reductions from integral representations, the Kampe de Feriet function, and elsewhere
单变量超几何函数pFq与薛定谔方程的幂级数解或量子力学中微扰展开的求和有关。对于这些应用，它是感兴趣的，以获得解析表达式，我们提出了一些情况下的pFp和p+1F_p，主要是p=2和p=3的减少。这些和相关的系列在量子和统计物理学和化学中有额外的应用。
Eigenproblem for Jacobi matrices: hypergeometric series solution
研究了d维一般三对角（Jacobi）矩阵的特征值和特征向量的摄动幂级数展开。（小）展开参数是长度为d-1的两条对角线与主对角线相交的条目，这给出了未扰动谱。该解决方案是明确的3d-5变量的多变量（霍恩型）超几何级数的一般情况下，或2d-3变量的特征值增长从一个角落矩阵元素。为了得到结果，我们首先重写的Jacobi矩阵的谱问题作为一个等效的三次方程组，然后通过应用多变量拉格朗日反演公式解决。相应的Jacobi行列式的计算显式。对于由特征向量分量组成的单项式，也给出了显式公式。
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