{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:UU2OIVEQDF7IS5PGUBEPRVJMH4","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":"867cd3ad6c64d210f5e35142c72a514eaf110fc931e44282addf7dff181de4df","cross_cats_sorted":["math-ph","math.CA","math.MP","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-11T15:53:41Z","title_canon_sha256":"b6768a2ebce01ef3febf88b8bd074f18b264a5293bd1bf21c00ebd0272c3d16f"},"schema_version":"1.0","source":{"id":"1307.3150","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.3150","created_at":"2026-05-18T03:18:07Z"},{"alias_kind":"arxiv_version","alias_value":"1307.3150v1","created_at":"2026-05-18T03:18:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.3150","created_at":"2026-05-18T03:18:07Z"},{"alias_kind":"pith_short_12","alias_value":"UU2OIVEQDF7I","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"UU2OIVEQDF7IS5PG","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"UU2OIVEQ","created_at":"2026-05-18T12:28:02Z"}],"graph_snapshots":[{"event_id":"sha256:1c5fc1c7a5ba570ea029ac46d8a62c4447e2534176346a908cdfabb51fe7ec32","target":"graph","created_at":"2026-05-18T03:18:07Z","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":"Jeffery's 1861 computations using finite difference calculus are resurrected and extended from forward differences to general delta operators and used to neatly prove theorems in the Rota--Mullins theory of polynomials of binomial type (Steffensen's poweroids) allowing, for example, compact treatments of umbral composition, the binomial property and the connection constants. It is shown that it forms a legitimate alternative to the usual umbral device and also anticipates a number of results obtained more recently.","authors_text":"J.S.Dowker","cross_cats":["math-ph","math.CA","math.MP","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-11T15:53:41Z","title":"Poweroids revisited - an old symbolic approach"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3150","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":"sha256:301d453b6c00ad613b15fe38f5490ea5343a5f2edfa0aa88e377425578495043","target":"record","created_at":"2026-05-18T03:18:07Z","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":"867cd3ad6c64d210f5e35142c72a514eaf110fc931e44282addf7dff181de4df","cross_cats_sorted":["math-ph","math.CA","math.MP","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-11T15:53:41Z","title_canon_sha256":"b6768a2ebce01ef3febf88b8bd074f18b264a5293bd1bf21c00ebd0272c3d16f"},"schema_version":"1.0","source":{"id":"1307.3150","kind":"arxiv","version":1}},"canonical_sha256":"a534e45490197e8975e6a048f8d52c3f29b69afbb3a3e27d1a8236aca118aa8e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a534e45490197e8975e6a048f8d52c3f29b69afbb3a3e27d1a8236aca118aa8e","first_computed_at":"2026-05-18T03:18:07.694243Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:07.694243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8w0KIEIB7jX+6XzRg4btrxNE90LJR0M+kG6VIVeEYm2aa6Y+TZdxXVQkdQrSjcwkupgzid7LcZCN3AuQI406Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:07.694874Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.3150","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:301d453b6c00ad613b15fe38f5490ea5343a5f2edfa0aa88e377425578495043","sha256:1c5fc1c7a5ba570ea029ac46d8a62c4447e2534176346a908cdfabb51fe7ec32"],"state_sha256":"4a07f252c88227946b3318a14ddd497649e537e3ad04d1c42cd61abe0c17c466"}