{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:ZLCMSK6J6EDF266XI232N52HE3","short_pith_number":"pith:ZLCMSK6J","canonical_record":{"source":{"id":"1907.08118","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-18T15:47:27Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"7645fc4d236165c93af75468895057fe194a0c79fbb6dbe2d263f02f03c5e191","abstract_canon_sha256":"673a5df1063be8a6d472b574854064d639fd12a528b5fa7d765249d1595d9b46"},"schema_version":"1.0"},"canonical_sha256":"cac4c92bc9f1065d7bd746b7a6f74726cc6aeb9700c19c64f64fa08c53134ce5","source":{"kind":"arxiv","id":"1907.08118","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.08118","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"arxiv_version","alias_value":"1907.08118v3","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.08118","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"pith_short_12","alias_value":"ZLCMSK6J6EDF","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"pith_short_16","alias_value":"ZLCMSK6J6EDF266X","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"pith_short_8","alias_value":"ZLCMSK6J","created_at":"2026-07-05T09:16:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:ZLCMSK6J6EDF266XI232N52HE3","target":"record","payload":{"canonical_record":{"source":{"id":"1907.08118","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-18T15:47:27Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"7645fc4d236165c93af75468895057fe194a0c79fbb6dbe2d263f02f03c5e191","abstract_canon_sha256":"673a5df1063be8a6d472b574854064d639fd12a528b5fa7d765249d1595d9b46"},"schema_version":"1.0"},"canonical_sha256":"cac4c92bc9f1065d7bd746b7a6f74726cc6aeb9700c19c64f64fa08c53134ce5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:16:42.409391Z","signature_b64":"Al50VWBHV74At5TkfmV2+fh77qUGbhCQ+m0oRoA2lAV5Bibrq60R+Nmb6jsOn8GC4mssKAORf2ZVXFztQKpbBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cac4c92bc9f1065d7bd746b7a6f74726cc6aeb9700c19c64f64fa08c53134ce5","last_reissued_at":"2026-07-05T09:16:42.408762Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:16:42.408762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.08118","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:16:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/UH+89nyLd0uD9LOwUEJDFtAy08r3XSTkKlVPG7Ge7qYVMLIfNZ2TiL3TEOW6g5L0OVdVPVUxR144/PXQXv3Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T22:54:51.312634Z"},"content_sha256":"9e3f4a96defb23b9b33b4317b886d103f3fae13289022747ad77d1f8de382430","schema_version":"1.0","event_id":"sha256:9e3f4a96defb23b9b33b4317b886d103f3fae13289022747ad77d1f8de382430"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:ZLCMSK6J6EDF266XI232N52HE3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A new trigonometric identity with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Hao Pan, Zhi-Wei Sun","submitted_at":"2019-07-18T15:47:27Z","abstract_excerpt":"In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$ we prove that $$\\sum_{k=1}^n(-1)^k(\\cot kx)\\sin k(n-k)x=\\frac{1-n}2,$$ which is equivalent to the identity $$\\sum_{k=1}^n(-1)^kU_{n-k}(\\cos kx)=-\\frac{n+1}2,$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$ we obtain $$\\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\\left(\\frac{n-k}2\\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.08118","kind":"arxiv","version":3},"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/1907.08118/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:16:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"49jzJPap6ettTaxY+p7vUPJJePeX+ImAX2N7+orOVs4cjSyNlVSJ+uj85kMNxwdWsruZwf0TwjEMOzsGJDSrAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T22:54:51.313545Z"},"content_sha256":"e9fa53033221d800924258d06c2d7c3bf61a3ac852268804fdefb94cb9df49f2","schema_version":"1.0","event_id":"sha256:e9fa53033221d800924258d06c2d7c3bf61a3ac852268804fdefb94cb9df49f2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3/bundle.json","state_url":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZLCMSK6J6EDF266XI232N52HE3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-03T22:54:51Z","links":{"resolver":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3","bundle":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3/bundle.json","state":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZLCMSK6J6EDF266XI232N52HE3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ZLCMSK6J6EDF266XI232N52HE3","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":"673a5df1063be8a6d472b574854064d639fd12a528b5fa7d765249d1595d9b46","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-18T15:47:27Z","title_canon_sha256":"7645fc4d236165c93af75468895057fe194a0c79fbb6dbe2d263f02f03c5e191"},"schema_version":"1.0","source":{"id":"1907.08118","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.08118","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"arxiv_version","alias_value":"1907.08118v3","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.08118","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"pith_short_12","alias_value":"ZLCMSK6J6EDF","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"pith_short_16","alias_value":"ZLCMSK6J6EDF266X","created_at":"2026-07-05T09:16:42Z"},{"alias_kind":"pith_short_8","alias_value":"ZLCMSK6J","created_at":"2026-07-05T09:16:42Z"}],"graph_snapshots":[{"event_id":"sha256:e9fa53033221d800924258d06c2d7c3bf61a3ac852268804fdefb94cb9df49f2","target":"graph","created_at":"2026-07-05T09:16:42Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1907.08118/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$ we prove that $$\\sum_{k=1}^n(-1)^k(\\cot kx)\\sin k(n-k)x=\\frac{1-n}2,$$ which is equivalent to the identity $$\\sum_{k=1}^n(-1)^kU_{n-k}(\\cos kx)=-\\frac{n+1}2,$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$ we obtain $$\\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\\left(\\frac{n-k}2\\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$.","authors_text":"Hao Pan, Zhi-Wei Sun","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-18T15:47:27Z","title":"A new trigonometric identity with applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.08118","kind":"arxiv","version":3},"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":"sha256:9e3f4a96defb23b9b33b4317b886d103f3fae13289022747ad77d1f8de382430","target":"record","created_at":"2026-07-05T09:16:42Z","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":"673a5df1063be8a6d472b574854064d639fd12a528b5fa7d765249d1595d9b46","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-18T15:47:27Z","title_canon_sha256":"7645fc4d236165c93af75468895057fe194a0c79fbb6dbe2d263f02f03c5e191"},"schema_version":"1.0","source":{"id":"1907.08118","kind":"arxiv","version":3}},"canonical_sha256":"cac4c92bc9f1065d7bd746b7a6f74726cc6aeb9700c19c64f64fa08c53134ce5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cac4c92bc9f1065d7bd746b7a6f74726cc6aeb9700c19c64f64fa08c53134ce5","first_computed_at":"2026-07-05T09:16:42.408762Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:16:42.408762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Al50VWBHV74At5TkfmV2+fh77qUGbhCQ+m0oRoA2lAV5Bibrq60R+Nmb6jsOn8GC4mssKAORf2ZVXFztQKpbBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:16:42.409391Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.08118","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9e3f4a96defb23b9b33b4317b886d103f3fae13289022747ad77d1f8de382430","sha256:e9fa53033221d800924258d06c2d7c3bf61a3ac852268804fdefb94cb9df49f2"],"state_sha256":"cb76378074264f10f712835fe59dd377e33a4e917a7e5be4ce7bdc7bd9f3df23"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RniO++SgZNDEB5OlmD7UxR/FeA7nZg0eCp32ZSeXNQ4rwL1+HE9Sx0HLigCs8eeaiYeo6a6pImgHPMPTdKKVDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T22:54:51.318952Z","bundle_sha256":"e2d1977772fd5eb02740fa3fa3e5f888e87e8132192d8042f3040eeaae0f3975"}}