{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:22WXJPPRRC63JGGTEUWTKJIWRW","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":"f378820779787186e40dcd2aacb027157faad4b5837edaa06a31163ed2c75dea","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-29T10:43:06Z","title_canon_sha256":"380b8c240e5fe052ff7eaa8136b52e45fb0fb56c3fce23a34ba3d2677f070a08"},"schema_version":"1.0","source":{"id":"1810.12048","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.12048","created_at":"2026-05-17T23:50:06Z"},{"alias_kind":"arxiv_version","alias_value":"1810.12048v2","created_at":"2026-05-17T23:50:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.12048","created_at":"2026-05-17T23:50:06Z"},{"alias_kind":"pith_short_12","alias_value":"22WXJPPRRC63","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"22WXJPPRRC63JGGT","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"22WXJPPR","created_at":"2026-05-18T12:31:59Z"}],"graph_snapshots":[{"event_id":"sha256:66ac783a23e1e95203c88c44e54a70df5244ff4d185737513cddf7efe00a503d","target":"graph","created_at":"2026-05-17T23:50:06Z","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":"We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem.","authors_text":"Alexander Berkovich, Ali K. Uncu","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-29T10:43:06Z","title":"Refined $q$-Trinomial Coefficients and Two Infinite Hierarchies of $q$-Series Identities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.12048","kind":"arxiv","version":2},"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":"sha256:60590905324414f89b723f2034926328aa1e8984a0c4304cddcd75a25d9fd793","target":"record","created_at":"2026-05-17T23:50:06Z","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":"f378820779787186e40dcd2aacb027157faad4b5837edaa06a31163ed2c75dea","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-29T10:43:06Z","title_canon_sha256":"380b8c240e5fe052ff7eaa8136b52e45fb0fb56c3fce23a34ba3d2677f070a08"},"schema_version":"1.0","source":{"id":"1810.12048","kind":"arxiv","version":2}},"canonical_sha256":"d6ad74bdf188bdb498d3252d3525168d867b04820ece1579f69de542def6b417","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6ad74bdf188bdb498d3252d3525168d867b04820ece1579f69de542def6b417","first_computed_at":"2026-05-17T23:50:06.836625Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:06.836625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bGAy2+f8NjGGJuHFfSCTh116K8M6QpU12hkBCh5/JMJnIzU9fqy6zQC5l73ETQuILE8ozyKIzs1SWzWIS0tzDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:06.837239Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.12048","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:60590905324414f89b723f2034926328aa1e8984a0c4304cddcd75a25d9fd793","sha256:66ac783a23e1e95203c88c44e54a70df5244ff4d185737513cddf7efe00a503d"],"state_sha256":"a96a0477089c86707a92754a9b1af4da409a749b1db88f53f04d378e51042add"}