In their paper on the gamma conjecture in mirror symmetry, Golyshev and Zagier introduce what we refer to as Frobenius constants associated to an ordinary linear differential operator L with a reflection type singularity. These numbers describe the variation around the reflection point of Frobenius solutions to L defined near other singular points. Golyshev and Zagier show that in certain geometric cases Frobenius constants are periods, and they raise the question quite generally how to describe these numbers motivically. In this paper we give a relation between Frobenius constants and Taylor coefficients of generalized gamma functions, from which it follows that Frobenius constants of Picard--Fuchs differential operators are periods. We also study the relation between these constants and periods of limiting Hodge structures. This is a major revision of the previous version of the manuscript. The notion of Frobenius constants and our main result are extended to the general case of regular singularities with any sets of local exponents. In addition, the generating function of Frobenius constants is given explicitly for all hypergeometric connections.

中文翻译：

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Gamma 函数、单一性和 Frobenius 常数

在他们关于镜像对称中伽马猜想的论文中，Golyshev 和 Zagier 介绍了我们所说的与具有反射型奇点的普通线性微分算子 L 相关的 Frobenius 常数。这些数字描述了在其他奇异点附近定义的 L 的 Frobenius 解的反射点周围的变化。Golyshev 和 Zagier 表明，在某些几何情况下，弗罗贝尼乌斯常数是周期，他们提出了如何从动机上描述这些数字的问题。本文给出了广义伽马函数的Frobenius常数和Taylor系数之间的关系，由此推导出Picard--Fuchs微分算子的Frobenius常数是周期。我们还研究了这些常数与限制霍奇结构周期之间的关系。这是对先前版本手稿的重大修订。Frobenius 常数的概念和我们的主要结果被扩展到具有任何局部指数集的规则奇点的一般情况。此外，还明确给出了所有超几何连接的 Frobenius 常数的生成函数。