{"paper":{"title":"A new extension of the Sun-Zagier result involving Bell numbers and derangement numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhi-Wei Sun","submitted_at":"2020-06-24T16:51:35Z","abstract_excerpt":"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 Stirli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.16089","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.16089/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}