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 Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {\tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {\tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp\'e de F\'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.
翻译:单级和多尺度的Feynman 集成体中的超强地理结构在广泛的表层类别中出现。 使用集成式关系, 必须计算相关的主元或斜体元件。 为此, 似乎有必要设计一种自动方法, 承认与相应更高超度函数相关的相( 部分) 差异方程式。 我们通过多值正规泰勒序列扩展系数的相关循环解决这些方程式。 在线性差异方程式的情况下, 扩展系数可以使用包 $t Sigma} 来决定。 在部分线性差异方程式的情况下, 也可以使用超度法方法来决定扩展系数。 在目前情况下, 会出现一种新的数量, 包括 Hurwitz 相形相形相形相形相形相形相色和通用的等式。 代码 HytmSypseries} 将差异方程式的类别转换成分析性序列扩展。 另外部分差异方程式, 具有合理解决方案和合理功能的波查默尔符号, 其中考虑使用代码 yt PartLDE 。 。 通用超地平地 函数 、 Apprical- ex- ex- ex- extra- exta- excial- exta- excial- excial- ex- ex- extipeal- exblection- exblection- ex- weal- weal- ex- sal- ex- sal- expal- sal- sal- sal- sliblementaldable- ex- ex- ex- ex- sal- sal- ex- ex- ex- exliction- extramental- sal- sal- exbal- expal- exp- slibal- sal- ex- sal- ex- ex- exption- exlibal- sal- sal- expal- sal- sal- sal- sal- sal- ex- ex- sal- sal- sal- sal- sal- sal- sal- sal- ex- ex- ex-