论文标题
Sun-Zagier结果的新扩展，涉及钟数和危险数字
A new extension of the Sun-Zagier result involving Bell numbers and derangement numbers
论文作者
论文摘要
让$ p $成为任何素数，让$ a $和$ n $为$ p \ nmid n $的正整数。我们表明$$ \ sum_ {k = 1}^{p^a-1} \ frac {b_k} {（ - n）是破坏数字。这扩展了Sun and Zagier于2011年出版的结果。此外，我们证明$$（-x）^n \ sum_ {k = 1}^{p^a-1} \ frac {b_k（x）} - \ sum_ {r = 1}^ax^{p^r} \ sum_ {k = 0}^{n-1} \ frac {（n-1）！} {k！}（-x）^k \ pmod {p \ pm \ pm \ mathbb z_p [x]}，$ $ $ b_k（x）= \ sum_ {l = 0}^ks（k，l）x^l $是$ s（k，l）\（0 \ le l \ le l \ le k）$ s（k，l \ le k）$ sectirl $ k $的钟声，第二类的stirl n of the第二类，$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ thbb z_p $是所有$ p $ p $ p $ - p $ p $ - p $ - integersic integers。
Let $p$ be any prime and let $a$ and $n$ be positive integers with $p\nmid n$. We show that $$\sum_{k=1}^{p^a-1}\frac{B_k}{(-n)^k}\equiv a(-1)^{n-1}D_{n-1}\pmod {p},$$ where $B_0,B_1,\ldots$ are the Bell numbers and $D_0,D_1,\ldots$ are the derangement numbers. This extends a result of Sun and Zagier published in 2011. Furthermore, we prove that $$(-x)^n\sum_{k=1}^{p^a-1}\frac{B_k(x)}{(-n)^k}\equiv -\sum_{r=1}^ax^{p^r}\sum_{k=0}^{n-1}\frac{(n-1)!}{k!}(-x)^k\pmod{p\mathbb Z_p[x]},$$ where $B_k(x)=\sum_{l=0}^kS(k,l)x^l$ is the Bell polynomial of degree $k$ with $S(k,l)\ (0\le l\le k)$ the Stirling numbers of the second kind, and $\mathbb Z_p$ is the ring of all $p$-adic integers.
