Abstract
We define and study a \(p\)-adic analogue of the incomplete gamma function related to Morita’s \(p\)-adic gamma function. We also discuss a combinatorial identity related to the Artin-Hasse series, which is a special case of the exponential principle in combinatorics. From this we deduce a curious \(p\)-adic property of \(\#\mathrm{Hom}(G,S_n)\) for a topologically finitely generated group \(G\), using a characterization of \(p\)-adic continuity for certain functions \(f \colon \mathbb Z_{>0} \to \mathbb Q_p\) due to O’Desky-Richman. In the end, we give an exposition of some standard properties of the Artin-Hasse series.
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Acknowledgments
This paper is the end product of an Illinois Geometry Lab project (an undergraduate research project) conducted by the third author (S.T.) in Spring 2021. IGL research is supported by the Department of Mathematics at the University of Illinois at Urbana-Champaign. We would like to express deep gratitude to Ravi Donepuli, who was the Graduate Assistant of this project, for his constant support throughout the semester, without which the project would not have been carried out smoothly and successfully. S.T. would like to thank Andy O’Desky for helpful comments on an earlier version of this draft, and we thank the anonymous referee for comments and corrections that helped us improve the exposition.
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Li, X., Reiter, J., Tang, S. et al. \(p\)-Adic Incomplete Gamma Functions and Artin-Hasse-Type Series. P-Adic Num Ultrametr Anal Appl 14, 335–343 (2022). https://doi.org/10.1134/S2070046622040070
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DOI: https://doi.org/10.1134/S2070046622040070