{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:UUGFSNOZMM6HG2UH3CA47GWE24","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":"99b0dcea94fc20ec0e87ab073e713074388c39a37412d16f5ce6edf83a0f49d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-10T18:32:57Z","title_canon_sha256":"d642021ff6db0e3153079b11a19456994361217f5331426b024ba7fd4c414f83"},"schema_version":"1.0","source":{"id":"1305.2404","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.2404","created_at":"2026-05-18T03:26:00Z"},{"alias_kind":"arxiv_version","alias_value":"1305.2404v1","created_at":"2026-05-18T03:26:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2404","created_at":"2026-05-18T03:26:00Z"},{"alias_kind":"pith_short_12","alias_value":"UUGFSNOZMM6H","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"UUGFSNOZMM6HG2UH","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"UUGFSNOZ","created_at":"2026-05-18T12:28:02Z"}],"graph_snapshots":[{"event_id":"sha256:18cf622b6a16bb30f86cbca3f9b5af1d0b7288fdc1cdc66287052b1f1b5a118f","target":"graph","created_at":"2026-05-18T03:26:00Z","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":"Let $H_n(t)$ denote the classical Rogers-Szeg\\\"o polynomial, and let $\\tH_n(t_1, \\ldots, t_l)$ denote the homogeneous Rogers-Szeg\\\"o polynomial in $l$ variables, with indeterminate $q$. There is a classical product formula for $H_k(t)H_n(t)$ as a sum of Rogers-Szeg\\\"o polynomials with coefficients being polynomials in $q$. We generalize this to a product formula for the multivariate homogeneous polynomials $\\tH_n(t_1, \\ldots, t_l)$. The coefficients given in the product formula are polynomials in $q$ which are defined recursively, and we find closed formulas for several interesting cases. We t","authors_text":"C. Ryan Vinroot, Stephen Cameron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-10T18:32:57Z","title":"A product formula for multivariate Rogers-Szeg\\\"o polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2404","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:cf3d08512c6a855b7499e2d745085a70c6c34d65afd34f50451dbcd86b2e3746","target":"record","created_at":"2026-05-18T03:26:00Z","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":"99b0dcea94fc20ec0e87ab073e713074388c39a37412d16f5ce6edf83a0f49d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-10T18:32:57Z","title_canon_sha256":"d642021ff6db0e3153079b11a19456994361217f5331426b024ba7fd4c414f83"},"schema_version":"1.0","source":{"id":"1305.2404","kind":"arxiv","version":1}},"canonical_sha256":"a50c5935d9633c736a87d881cf9ac4d7316deebcec3da60aa37ea44ebdd02534","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a50c5935d9633c736a87d881cf9ac4d7316deebcec3da60aa37ea44ebdd02534","first_computed_at":"2026-05-18T03:26:00.087154Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:00.087154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I4SB7rxUzKj5UhZaNAz2lZ4XwiTi9DUW1LXPXt4iJEb+xs7DTJk4afl17WA6+t8Mx+vqJGGcMs/sxzBWD3WZBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:00.087889Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.2404","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cf3d08512c6a855b7499e2d745085a70c6c34d65afd34f50451dbcd86b2e3746","sha256:18cf622b6a16bb30f86cbca3f9b5af1d0b7288fdc1cdc66287052b1f1b5a118f"],"state_sha256":"63357f997bf32a26c53836d6fd24e0496acb2311b08185b4f5cec2cf628cc9a0"}