Derangements and the $p$-adic incomplete gamma function

关于 $p$-adic 不完整 Mellin 变换和 $p$-adic 不完整 gamma 函数

On $p$-adic incomplete Mellin transforms and $p$-adic incomplete gamma-functions

本文提出了一种在更弱条件下构造 $p$-adic 不完整 gamma 函数的方法，利用 $p$-adic 积分变换，适用于所有除了有限多个素数外的素数。

P-adic incomplete gamma functions and Artin-Hasse-type series

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 Arti…

Factorial-Type Recurrence Relations and $p$-adic Incomplete Gamma Functions

We introduce an automorphism $\mathcal{S}$ of the space $C(\mathbb{Z}_p,\mathbb{C}_p)$ of continuous functions $\mathbb{Z}_p \rightarrow \mathbb{C}_p$ and show that it can be used to give an alternative construction of the $p$-adic incomplete $Γ$-functions recently introduced by O'Desky and Richman (arXiv:2012.04615). We then describe various properties of the automorphism $\mathcal{S}$, showing that it is self-adjoint with respect to a certain non-degenerate symmetric bilinear form defined in terms of $p$-adic integration, and showing that its inverse plays a role in a $p$-adic integral-trans…

一类不完全特征和

A Class of Incomplete Character Sums

本文借助光滑层的张量诱导的ℓ-adic上同调方法，研究了一类不完全特征和的性质。