{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CZ7ARRFIPDPBRJBSQ5342YDTFA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f2bc5dac94c7f87582842966b21f84a0fc6ee52505da5a1de522fad813a0cfb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-09-18T21:35:59Z","title_canon_sha256":"508c6e60ae21e097873327911db41ef8ac9a2a83a1eae682c19e8b51234bd652"},"schema_version":"1.0","source":{"id":"1209.4110","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4110","created_at":"2026-05-18T03:45:21Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4110v1","created_at":"2026-05-18T03:45:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4110","created_at":"2026-05-18T03:45:21Z"},{"alias_kind":"pith_short_12","alias_value":"CZ7ARRFIPDPB","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CZ7ARRFIPDPBRJBS","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CZ7ARRFI","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:b305404b0f302aed8d3b94da20d5cd1b0ebcf7340163dc223caba6b7b18377eb","target":"graph","created_at":"2026-05-18T03:45:21Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The modified\nB_{n}^{*} = \\sum_{r=0}^{n} \\binom{n+r}{2r} \\frac{B_{r}}{n+r}, \\quad n > 0\nintroduced by D. Zagier in 1998 are extended to the polynomial case by replacing $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Properties of these new polynomials are established using the umbral method as well as classical techniques. The values of $x$ that yield periodic subsequences $B_{2n+1}^{*}(x)$ are classified. The strange 6-periodicity of $B_{2n+1}^{*}$, established by Zagier, is explained by exhibiting a decomposition of this sequence as the sum of two parts with periods 2 and 3, respectively. S","authors_text":"Atul Dixit, Christophe Vignat, Victor H. Moll","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-09-18T21:35:59Z","title":"The Zagier modification of Bernoulli numbers and a polynomial extension. Part I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4110","kind":"arxiv","version":1},"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:db52a3f54c8aba71556faf94115bfde10cf1a35820a537a0856b669213cf80eb","target":"record","created_at":"2026-05-18T03:45:21Z","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":"f2bc5dac94c7f87582842966b21f84a0fc6ee52505da5a1de522fad813a0cfb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-09-18T21:35:59Z","title_canon_sha256":"508c6e60ae21e097873327911db41ef8ac9a2a83a1eae682c19e8b51234bd652"},"schema_version":"1.0","source":{"id":"1209.4110","kind":"arxiv","version":1}},"canonical_sha256":"167e08c4a878de18a4328777cd6073280fb558767195a3f1fa1e72d4c54a6c7c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"167e08c4a878de18a4328777cd6073280fb558767195a3f1fa1e72d4c54a6c7c","first_computed_at":"2026-05-18T03:45:21.561874Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:21.561874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CvVJR5crRNFbjC4Cd3aeEEJChzT09mV9GGheM10/ef+axK2zSG1IqfH6UjzUubLh8OYr8ZQ6TGY3neOKgzzcAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:21.562547Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.4110","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db52a3f54c8aba71556faf94115bfde10cf1a35820a537a0856b669213cf80eb","sha256:b305404b0f302aed8d3b94da20d5cd1b0ebcf7340163dc223caba6b7b18377eb"],"state_sha256":"ef0b3a523f5a6bdc0bad2e9cce274b2d5329fadbb87c3ea128d6f4f70c5f8d3f"}