我们怎么能摆脱阿惠的关系呢？
How can we escape Thomae's relations?
摘要：
在1879年，Rumiae讨论了两个一般的超几何$_3F_2$ -系列与参数1之间的关系。众所周知，自那时以来，有120个这样的关系（包括平凡的那些来自排列的参数的超几何级数）。最近，莱茵和维奥拉提出了以下问题（在不同的，但等价的语言积分）：如果存在一个线性依赖关系超过$\mathbf Q$之间的两个收敛$_3F_2$ -系列的参数1，与积分参数，其价值观是无理数，是这种关系的专业化之一的120个关系？几年后，佐藤回答了这个问题的负面影响，给出了六个例子的关系，不能解释的关系。我们表明，Sato的反例可以自然地嵌入到两个无限多个$_3F_2$关系族中，两者都由三个独立参数参数化。此外，我们还发现了两个性质相同的无限族。族，这似乎没有被记录之前，来自某些$_3F_2$ -转换公式和连续关系。详细地说明了Rhin积分和Viola积分与$_3F_2$ -级数的关系。
英文摘要：
In 1879, Thomae discussed the relations between two generic hypergeometric $_3F_2$-series with argument 1. It is well-known since then that there are 120 such relations (including the trivial ones which come from permutations of the parameters of the hypergeometric series). More recently, Rhin and Viola asked the following question (in a different, but equivalent language of integrals): If there exists a linear dependence relation over $\mathbf Q$ between two convergent $_3F_2$-series with argument 1, with integral parameters, and whose values are irrational numbers, is this relation a specialisation of one of the 120 Thomae relations? A few years later, Sato answered this question in the negative, by giving six examples of relations which cannot be explained by Thomae's relations. We show that Sato's counter-examples can be naturally embedded into two families of infinitely many $_3F_2$-relations, both parametrised by three independent parameters. Moreover, we find two more infinite families of the same nature. The families, which do not seem to have been recorded before, come from certain $_3F_2$-transformation formulae and contiguous relations. We also explain in detail the relationship between the integrals of Rhin and Viola and $_3F_2$-series.
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