{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:RXT5IJB6WT5HSRMCTN6KSYKT2Y","short_pith_number":"pith:RXT5IJB6","schema_version":"1.0","canonical_sha256":"8de7d4243eb4fa7945829b7ca96153d61d4ab4ab1df0467b7c52c2082238d382","source":{"kind":"arxiv","id":"2401.14197","version":1},"attestation_state":"computed","paper":{"title":"Proof of conjectures on series with summands involving $ \\binom{2k}{k}8^k/(\\binom{3k}{k}\\binom{6k}{3k})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Yajun Zhou, Zhi-Wei Sun","submitted_at":"2024-01-25T14:18:15Z","abstract_excerpt":"Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\\binom{2k}k8^k/(\\binom{3k}k\\binom{6k}{3k})$. For example, we prove that \\[\\sum_{k=0}^\\infty\\frac{(350k-17)\\binom{2k}k8^k} {\\binom{3k}k\\binom{6k}{3k}}=15\\sqrt2\\,\\pi+27\\] and \\[\\sum_{k=1}^\\infty\\frac{\\left\\{(5k-1)\\left[16\\mathsf H_{2k-1}^{(2)}-3\\mathsf H_{k-1}^{(2)}\\right]-\\frac{12(6k-1)}{(2k-1)^2}\\right\\}\\binom{2k}k8^k} {k(2k-1)\\binom{3k}k\\binom{6k}{3k}}=\\frac{\\pi^3}{12\\sqrt2},\\] where $\\mathsf H^{(2)}_m$ denotes the second-order harmonic numbe"},"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":"2401.14197","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2024-01-25T14:18:15Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"a9f370aa7d32dc1cbae1952fb60c3c185bf51ecd8339e2284357c91d7e59272d","abstract_canon_sha256":"c9be6c74f24746d0f6559072c6e9c476e009e6fc2695729941193b861f61694c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:37:32.597978Z","signature_b64":"yNNOmfyJcwdmhlngT4vGe5TtbcDzkr4kdoZjm/bWXYaiBKvolr2yikWRAXWnBKhaLfWg/OQFWFzmZD29iAODCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8de7d4243eb4fa7945829b7ca96153d61d4ab4ab1df0467b7c52c2082238d382","last_reissued_at":"2026-07-05T07:37:32.597528Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:37:32.597528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of conjectures on series with summands involving $ \\binom{2k}{k}8^k/(\\binom{3k}{k}\\binom{6k}{3k})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Yajun Zhou, Zhi-Wei Sun","submitted_at":"2024-01-25T14:18:15Z","abstract_excerpt":"Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\\binom{2k}k8^k/(\\binom{3k}k\\binom{6k}{3k})$. For example, we prove that \\[\\sum_{k=0}^\\infty\\frac{(350k-17)\\binom{2k}k8^k} {\\binom{3k}k\\binom{6k}{3k}}=15\\sqrt2\\,\\pi+27\\] and \\[\\sum_{k=1}^\\infty\\frac{\\left\\{(5k-1)\\left[16\\mathsf H_{2k-1}^{(2)}-3\\mathsf H_{k-1}^{(2)}\\right]-\\frac{12(6k-1)}{(2k-1)^2}\\right\\}\\binom{2k}k8^k} {k(2k-1)\\binom{3k}k\\binom{6k}{3k}}=\\frac{\\pi^3}{12\\sqrt2},\\] where $\\mathsf H^{(2)}_m$ denotes the second-order harmonic numbe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.14197","kind":"arxiv","version":1},"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/2401.14197/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":"2401.14197","created_at":"2026-07-05T07:37:32.597583+00:00"},{"alias_kind":"arxiv_version","alias_value":"2401.14197v1","created_at":"2026-07-05T07:37:32.597583+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.14197","created_at":"2026-07-05T07:37:32.597583+00:00"},{"alias_kind":"pith_short_12","alias_value":"RXT5IJB6WT5H","created_at":"2026-07-05T07:37:32.597583+00:00"},{"alias_kind":"pith_short_16","alias_value":"RXT5IJB6WT5HSRMC","created_at":"2026-07-05T07:37:32.597583+00:00"},{"alias_kind":"pith_short_8","alias_value":"RXT5IJB6","created_at":"2026-07-05T07:37:32.597583+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.24588","citing_title":"Functional Dilogarithm Identities in Quadratic Fields","ref_index":40,"is_internal_anchor":false},{"citing_arxiv_id":"2604.24588","citing_title":"Functional Dilogarithm Identities in Quadratic Fields","ref_index":40,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RXT5IJB6WT5HSRMCTN6KSYKT2Y","json":"https://pith.science/pith/RXT5IJB6WT5HSRMCTN6KSYKT2Y.json","graph_json":"https://pith.science/api/pith-number/RXT5IJB6WT5HSRMCTN6KSYKT2Y/graph.json","events_json":"https://pith.science/api/pith-number/RXT5IJB6WT5HSRMCTN6KSYKT2Y/events.json","paper":"https://pith.science/paper/RXT5IJB6"},"agent_actions":{"view_html":"https://pith.science/pith/RXT5IJB6WT5HSRMCTN6KSYKT2Y","download_json":"https://pith.science/pith/RXT5IJB6WT5HSRMCTN6KSYKT2Y.json","view_paper":"https://pith.science/paper/RXT5IJB6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2401.14197&json=true","fetch_graph":"https://pith.science/api/pith-number/RXT5IJB6WT5HSRMCTN6KSYKT2Y/graph.json","fetch_events":"https://pith.science/api/pith-number/RXT5IJB6WT5HSRMCTN6KSYKT2Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RXT5IJB6WT5HSRMCTN6KSYKT2Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RXT5IJB6WT5HSRMCTN6KSYKT2Y/action/storage_attestation","attest_author":"https://pith.science/pith/RXT5IJB6WT5HSRMCTN6KSYKT2Y/action/author_attestation","sign_citation":"https://pith.science/pith/RXT5IJB6WT5HSRMCTN6KSYKT2Y/action/citation_signature","submit_replication":"https://pith.science/pith/RXT5IJB6WT5HSRMCTN6KSYKT2Y/action/replication_record"}},"created_at":"2026-07-05T07:37:32.597583+00:00","updated_at":"2026-07-05T07:37:32.597583+00:00"}