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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 $j$."},"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":"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"},"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"},"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1907.08118","created_at":"2026-07-05T09:16:42.408831+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.08118v3","created_at":"2026-07-05T09:16:42.408831+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.08118","created_at":"2026-07-05T09:16:42.408831+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZLCMSK6J6EDF","created_at":"2026-07-05T09:16:42.408831+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZLCMSK6J6EDF266X","created_at":"2026-07-05T09:16:42.408831+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZLCMSK6J","created_at":"2026-07-05T09:16:42.408831+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/ZLCMSK6J6EDF266XI232N52HE3","json":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3.json","graph_json":"https://pith.science/api/pith-number/ZLCMSK6J6EDF266XI232N52HE3/graph.json","events_json":"https://pith.science/api/pith-number/ZLCMSK6J6EDF266XI232N52HE3/events.json","paper":"https://pith.science/paper/ZLCMSK6J"},"agent_actions":{"view_html":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3","download_json":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3.json","view_paper":"https://pith.science/paper/ZLCMSK6J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.08118&json=true","fetch_graph":"https://pith.science/api/pith-number/ZLCMSK6J6EDF266XI232N52HE3/graph.json","fetch_events":"https://pith.science/api/pith-number/ZLCMSK6J6EDF266XI232N52HE3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3/action/storage_attestation","attest_author":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3/action/author_attestation","sign_citation":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3/action/citation_signature","submit_replication":"https://pith.science/pith/ZLCMSK6J6EDF266XI232N52HE3/action/replication_record"}},"created_at":"2026-07-05T09:16:42.408831+00:00","updated_at":"2026-07-05T09:16:42.408831+00:00"}