{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:4B75BOANURDUZXYVAPOSFWNNA3","short_pith_number":"pith:4B75BOAN","canonical_record":{"source":{"id":"1702.02067","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-07T15:42:24Z","cross_cats_sorted":[],"title_canon_sha256":"8ca78a89292c93b9105d5734982d1dd6d02e1b8be9e1c8dc589bb04ce1aab2b8","abstract_canon_sha256":"1ccd6e8ec8738225c7b2afd5640583cd1084fc396a23ad3fb10fe969690f2d2b"},"schema_version":"1.0"},"canonical_sha256":"e07fd0b80da4474cdf1503dd22d9ad06f9b11cfcfbb0a3a0cb77e2de0d74c05f","source":{"kind":"arxiv","id":"1702.02067","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02067","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02067v1","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02067","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"pith_short_12","alias_value":"4B75BOANURDU","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"4B75BOANURDUZXYV","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"4B75BOAN","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:4B75BOANURDUZXYVAPOSFWNNA3","target":"record","payload":{"canonical_record":{"source":{"id":"1702.02067","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-07T15:42:24Z","cross_cats_sorted":[],"title_canon_sha256":"8ca78a89292c93b9105d5734982d1dd6d02e1b8be9e1c8dc589bb04ce1aab2b8","abstract_canon_sha256":"1ccd6e8ec8738225c7b2afd5640583cd1084fc396a23ad3fb10fe969690f2d2b"},"schema_version":"1.0"},"canonical_sha256":"e07fd0b80da4474cdf1503dd22d9ad06f9b11cfcfbb0a3a0cb77e2de0d74c05f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:23.904343Z","signature_b64":"UgzEqT5hQUfasNi+qTDuO8CemQRxiwo0vv+6BkagBrADFRuirYiCBoD1bveGz3eyitkMzUOstl7gPzDQjFXwBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e07fd0b80da4474cdf1503dd22d9ad06f9b11cfcfbb0a3a0cb77e2de0d74c05f","last_reissued_at":"2026-05-18T00:11:23.903815Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:23.903815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1702.02067","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:11:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JsHRNU03vZodVyA0n8pDj6Nu1rm9XIES+PSQEDbj+uOgZXCg2yKiWzxm7qniQArgdau29u+YjbOsWyirE/dzCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:33:01.534085Z"},"content_sha256":"97c7f1739249dd0d01725e30b24bad2130aa7ab7d3bbfc807a1fca2cdc31e67b","schema_version":"1.0","event_id":"sha256:97c7f1739249dd0d01725e30b24bad2130aa7ab7d3bbfc807a1fca2cdc31e67b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:4B75BOANURDUZXYVAPOSFWNNA3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A $q$-analogue of $\\bar\\alpha$-Whitney Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"B. S. El-Desouky, F. A. Shiha","submitted_at":"2017-02-07T15:42:24Z","abstract_excerpt":"We define the $(q,\\bar{\\boldsymbol{\\alpha}})$-Whitney numbers which are reduced to the $\\bar{\\boldsymbol{\\alpha}}$-Whitney numbers when $q\\rightarrow1$. Moreover, we obtain several properties of these numbers such as explicit formulas, recurrence relations, generating functions, orthogonality and inverse relations. Finally, we define the $\\bar{\\boldsymbol{\\alpha}}$-Whitney-Lah numbers as a generalization of the $r$-Whitney-Lah numbers and we introduce their important basic properties."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02067","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:11:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y3ziLgbs7MxnFYYgGeA371oA3+27wNvWjGEmNpzP8fydWunH7wIZC8PFSJNlAEDKJg0UeJl2wqkRDL0eTG/7DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:33:01.534450Z"},"content_sha256":"c68ec3bfd189c70c9065650542674b68606136a904ef3c7e22102a610faa2d3a","schema_version":"1.0","event_id":"sha256:c68ec3bfd189c70c9065650542674b68606136a904ef3c7e22102a610faa2d3a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4B75BOANURDUZXYVAPOSFWNNA3/bundle.json","state_url":"https://pith.science/pith/4B75BOANURDUZXYVAPOSFWNNA3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4B75BOANURDUZXYVAPOSFWNNA3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-26T13:33:01Z","links":{"resolver":"https://pith.science/pith/4B75BOANURDUZXYVAPOSFWNNA3","bundle":"https://pith.science/pith/4B75BOANURDUZXYVAPOSFWNNA3/bundle.json","state":"https://pith.science/pith/4B75BOANURDUZXYVAPOSFWNNA3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4B75BOANURDUZXYVAPOSFWNNA3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:4B75BOANURDUZXYVAPOSFWNNA3","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":"1ccd6e8ec8738225c7b2afd5640583cd1084fc396a23ad3fb10fe969690f2d2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-07T15:42:24Z","title_canon_sha256":"8ca78a89292c93b9105d5734982d1dd6d02e1b8be9e1c8dc589bb04ce1aab2b8"},"schema_version":"1.0","source":{"id":"1702.02067","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02067","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02067v1","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02067","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"pith_short_12","alias_value":"4B75BOANURDU","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"4B75BOANURDUZXYV","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"4B75BOAN","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:c68ec3bfd189c70c9065650542674b68606136a904ef3c7e22102a610faa2d3a","target":"graph","created_at":"2026-05-18T00:11:23Z","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 define the $(q,\\bar{\\boldsymbol{\\alpha}})$-Whitney numbers which are reduced to the $\\bar{\\boldsymbol{\\alpha}}$-Whitney numbers when $q\\rightarrow1$. Moreover, we obtain several properties of these numbers such as explicit formulas, recurrence relations, generating functions, orthogonality and inverse relations. Finally, we define the $\\bar{\\boldsymbol{\\alpha}}$-Whitney-Lah numbers as a generalization of the $r$-Whitney-Lah numbers and we introduce their important basic properties.","authors_text":"B. S. El-Desouky, F. A. Shiha","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-07T15:42:24Z","title":"A $q$-analogue of $\\bar\\alpha$-Whitney Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02067","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":"sha256:97c7f1739249dd0d01725e30b24bad2130aa7ab7d3bbfc807a1fca2cdc31e67b","target":"record","created_at":"2026-05-18T00:11:23Z","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":"1ccd6e8ec8738225c7b2afd5640583cd1084fc396a23ad3fb10fe969690f2d2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-07T15:42:24Z","title_canon_sha256":"8ca78a89292c93b9105d5734982d1dd6d02e1b8be9e1c8dc589bb04ce1aab2b8"},"schema_version":"1.0","source":{"id":"1702.02067","kind":"arxiv","version":1}},"canonical_sha256":"e07fd0b80da4474cdf1503dd22d9ad06f9b11cfcfbb0a3a0cb77e2de0d74c05f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e07fd0b80da4474cdf1503dd22d9ad06f9b11cfcfbb0a3a0cb77e2de0d74c05f","first_computed_at":"2026-05-18T00:11:23.903815Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:11:23.903815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UgzEqT5hQUfasNi+qTDuO8CemQRxiwo0vv+6BkagBrADFRuirYiCBoD1bveGz3eyitkMzUOstl7gPzDQjFXwBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:11:23.904343Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.02067","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:97c7f1739249dd0d01725e30b24bad2130aa7ab7d3bbfc807a1fca2cdc31e67b","sha256:c68ec3bfd189c70c9065650542674b68606136a904ef3c7e22102a610faa2d3a"],"state_sha256":"5fafec3e701b83043e6970bb6f8217e8e78754bae1d5ea2fdce6f94180661006"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MIL+ONniJ6M/7/cxbyExIf+Tl57xhq77kZm1kHZR8TvAcnt5rG0VLV8ftEfGiL0A3Telpn7LWVhsX/FVx3Z6DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T13:33:01.536426Z","bundle_sha256":"4b881233370061ca3b9eb737e12e57c00ce8190292fee07dd47637ecd768af93"}}