In this paper, we use combinatorial group theory and a limiting process to connect various types of hypergeometric series, and of relations among such series. We begin with a set S of 56 distinct translates of a certain function M, which takes the form of a Barnes integral, and is expressible as a sum of two very-well-poised \(_9F_8\) hypergeometric series of unit argument. We consider a known, transitive action of the Coxeter group \(W(E_7)\) on this set. We show that, by removing from \(W(E_7)\) a particular generator, we obtain a subgroup that is isomorphic to \(W(D_6)\), and that acts intransitively on S, partitioning it into three orbits, of sizes 32, 12, and 12, respectively. Taking certain limits of the M functions in the first orbit yields a set of 32 J functions, each of which is a sum of two Saalschützian \(_4F_3\) hypergeometric series of unit argument. The original action of \(W(D_6)\) on the M functions in this orbit is then seen to correspond to a known action of this group on this set of J functions. In a similar way, the image of each of the size-12 orbits, under a similar limiting process, is a set of 12 L functions that have been investigated in earlier works. In fact, these two image sets are the same. The limiting process is seen to preserve distance, except on pairs consisting of one M function from each size-12 orbit. Finally, each known three-term relation among the J and L functions is seen to be obtainable as a limit of a known three-term relation among the M functions.

在本文中，我们使用组合群论和一个限制过程来连接各种类型的超几何级数，以及这些级数之间的关系。我们从某个函数M的56个不同平移的集合S开始，该函数采用Barnes积分的形式，并且可以表示为两个非常均衡的\（_ 9F_8 \）超几何级数单位论证的和。我们考虑Coxeter组\（W（E_7）\）在此集合上的已知传递动作。我们显示出，通过从\（W（E_7）\）中删除特定生成器，我们获得了与\（W（D_6）\）同构的子组，该子组对S进行不传递，将其分成三个分别为32、12和12的轨道。对第一个轨道上的M个函数取一定的限制会产生一组32个J函数，每个J函数都是两个Saalschützian \（_ 4F_3 \）超几何级数单位论证的和。然后，可以看到\（W（D_6）\）在此轨道上的M个函数上的原始动作对应于该组在这组J函数上的已知动作。以类似的方式，在类似的限制过程中，每个12尺寸轨道的图像都是一组12 L在早期作品中已经研究过的功能。实际上，这两个图像集是相同的。限制过程被认为可以保持距离，除了每个12号轨道上由一个M函数组成的对。最后，J和L函数之间的每个已知三项关系被视为可获得，作为M函数之间已知的三项关系的极限。