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<p>This paper introduces a <i>p</i>-adic analogue of Gauss’s hypergeometric function, constructed via a method that is distinct from Dwork’s approach. The idea of our construction is motivated by the Ohno-Zagier formula, which is elucidated through the relationship between the hypergeometric differential equation and the Knizhnik-Zamolodchikov (KZ) equation. We develop a rigorous framework for the residue-wise analytic prolongation of our <i>p</i>-adic hypergeometric function by exploring its relationship with <i>p</i>-adic multiple polylogarithms. Through a detailed analysis of its local behavior near the point 1, we show a <i>p</i>-adic version of Gauss hypergeometric theorem for the function.</p>

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p-Adic Hypergeometric Function Related with p-Adic Multiple Polylogarithms

  • 期刊论文

  • 发表日期：

    2026年5月23日

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  • Hidekazu Furusho

摘要

This paper introduces a p-adic analogue of Gauss’s hypergeometric function, constructed via a method that is distinct from Dwork’s approach. The idea of our construction is motivated by the Ohno-Zagier formula, which is elucidated through the relationship between the hypergeometric differential equation and the Knizhnik-Zamolodchikov (KZ) equation. We develop a rigorous framework for the residue-wise analytic prolongation of our p-adic hypergeometric function by exploring its relationship with p-adic multiple polylogarithms. Through a detailed analysis of its local behavior near the point 1, we show a p-adic version of Gauss hypergeometric theorem for the function.