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arXiv · 2012.04615

Derangements and the $p$-adic incomplete gamma function

Andrew O'Desky·David Harry Richman
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Abstract

We introduce a $p$-adic analogue of the incomplete gamma function. We also introduce quantities ($m$-values) associated to a function on natural numbers and prove a new characterization of $p$-adic continuity for functions with $p$-integral $m$-values. Combinatorial interpretations for the integral values of the incomplete gamma function and functions with $m$-values zero or one are obtained, which show that these functions count derangements in generalized symmetric groups and permutations with restricted cycle lengths.

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Andrew O'Desky, David Harry Richman. 2020-12-08. Derangements and the $p$-adic incomplete gamma function. https://doi.org/10.1090/tran/8716

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