Title: Derangements and the $p$-adic incomplete gamma function Show Chinese title

Title: 错位排列和$p$-进制不完全伽马函数

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. Show Chinese abstract

Abstract: 我们引入了不完全伽马函数的$p$-adic 类比。 我们还引入了与自然数上的函数相关的量（$m$-值），并证明了具有$p$-积分$m$-值的函数的$p$-adic 连续性的新特征。 得到了不完全伽马函数和具有$m$-值为零或一的函数的积分值的组合解释，这些解释表明这些函数在广义对称群中计数错位排列以及具有限制循环长度的排列。