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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","authors_text":"Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-24T16:51:35Z","title":"A new extension of the Sun-Zagier result involving Bell numbers and derangement numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.16089","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:76363e8415fc7a33ec50bb5ad0a8e5b041bff7b8d005a4c6929632d07106ee67","target":"record","created_at":"2026-07-05T01:45:19Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a78a87ce17a1373fcb075da80591886651998a8c2a577fde6fcb207b92f4e36d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-24T16:51:35Z","title_canon_sha256":"31dbf94d1e5ea5149fa3f55cb48a402ff8b74ba68428b7cf818656824778b2cc"},"schema_version":"1.0","source":{"id":"2006.16089","kind":"arxiv","version":2}},"canonical_sha256":"ecb2a5b7e3db0f9bcba29b2572fbd30b7dbf0ab8cf2f3d2841cf4012d36ece22","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ecb2a5b7e3db0f9bcba29b2572fbd30b7dbf0ab8cf2f3d2841cf4012d36ece22","first_computed_at":"2026-07-05T01:45:19.988502Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:45:19.988502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OnFJv+IM15LQc6mboKkkFwkSd+s1XBbZltAcuJxrIxTkpG5WzNyTwQRuwx364Tcivxo7OEZ6NrNBkWQpUBKIAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:45:19.988961Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.16089","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:76363e8415fc7a33ec50bb5ad0a8e5b041bff7b8d005a4c6929632d07106ee67","sha256:ad2afc5e4391948b673b6f4d8aeb154137fb90af0a0ed61172a8993c52d4c8d8"],"state_sha256":"2eb581fb75aba194769b24715dd513c45b027d775f4ac6b16b2d8d83ba9d3422"}