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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2006.16089","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-06-24T16:51:35Z","cross_cats_sorted":[],"title_canon_sha256":"31dbf94d1e5ea5149fa3f55cb48a402ff8b74ba68428b7cf818656824778b2cc","abstract_canon_sha256":"a78a87ce17a1373fcb075da80591886651998a8c2a577fde6fcb207b92f4e36d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:45:19.988961Z","signature_b64":"OnFJv+IM15LQc6mboKkkFwkSd+s1XBbZltAcuJxrIxTkpG5WzNyTwQRuwx364Tcivxo7OEZ6NrNBkWQpUBKIAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ecb2a5b7e3db0f9bcba29b2572fbd30b7dbf0ab8cf2f3d2841cf4012d36ece22","last_reissued_at":"2026-07-05T01:45:19.988502Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:45:19.988502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2006.16089","created_at":"2026-07-05T01:45:19.988561+00:00"},{"alias_kind":"arxiv_version","alias_value":"2006.16089v2","created_at":"2026-07-05T01:45:19.988561+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.16089","created_at":"2026-07-05T01:45:19.988561+00:00"},{"alias_kind":"pith_short_12","alias_value":"5SZKLN7D3MHZ","created_at":"2026-07-05T01:45:19.988561+00:00"},{"alias_kind":"pith_short_16","alias_value":"5SZKLN7D3MHZXS5C","created_at":"2026-07-05T01:45:19.988561+00:00"},{"alias_kind":"pith_short_8","alias_value":"5SZKLN7D","created_at":"2026-07-05T01:45:19.988561+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5SZKLN7D3MHZXS5CTMSXF66TBN","json":"https://pith.science/pith/5SZKLN7D3MHZXS5CTMSXF66TBN.json","graph_json":"https://pith.science/api/pith-number/5SZKLN7D3MHZXS5CTMSXF66TBN/graph.json","events_json":"https://pith.science/api/pith-number/5SZKLN7D3MHZXS5CTMSXF66TBN/events.json","paper":"https://pith.science/paper/5SZKLN7D"},"agent_actions":{"view_html":"https://pith.science/pith/5SZKLN7D3MHZXS5CTMSXF66TBN","download_json":"https://pith.science/pith/5SZKLN7D3MHZXS5CTMSXF66TBN.json","view_paper":"https://pith.science/paper/5SZKLN7D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2006.16089&json=true","fetch_graph":"https://pith.science/api/pith-number/5SZKLN7D3MHZXS5CTMSXF66TBN/graph.json","fetch_events":"https://pith.science/api/pith-number/5SZKLN7D3MHZXS5CTMSXF66TBN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5SZKLN7D3MHZXS5CTMSXF66TBN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5SZKLN7D3MHZXS5CTMSXF66TBN/action/storage_attestation","attest_author":"https://pith.science/pith/5SZKLN7D3MHZXS5CTMSXF66TBN/action/author_attestation","sign_citation":"https://pith.science/pith/5SZKLN7D3MHZXS5CTMSXF66TBN/action/citation_signature","submit_replication":"https://pith.science/pith/5SZKLN7D3MHZXS5CTMSXF66TBN/action/replication_record"}},"created_at":"2026-07-05T01:45:19.988561+00:00","updated_at":"2026-07-05T01:45:19.988561+00:00"}