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Congruence relations for p-adic hypergeometric functions \(\widehat{{\mathscr {F}}}_{a,...,a}^{(\sigma )}(t)\) and its transformation formula


We introduce new kinds of p-adic hypergeometric functions. We show these functions satisfy congruence relations similar to Dwork’s p-adic hypergeometric functions, so they are convergent functions. We also show that there is a transformation formula between our new p-adic hypergeometric functions and p-adic hypergeometric functions of logarithmic type defined in Asakura (New p-adic hypergeometric functions and syntomic regulators. arXiv:1811.03770) in a particular case.

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I truly appreciate the help of Professor Masanori Asakura. He gave the definition of our new p-adic hypergeometric functions and the conjectures of transformation formulas with the aid of computer. Also, he gave a lot of advice of this paper.

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Correspondence to Wang Chung-Hsuan.

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Chung-Hsuan, W. Congruence relations for p-adic hypergeometric functions \(\widehat{{\mathscr {F}}}_{a,...,a}^{(\sigma )}(t)\) and its transformation formula. manuscripta math. (2021).

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Mathematics Subject Classification

  • 33E50