{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:2IUVRPNRSOG6HHCUXTFOLKLKEB","short_pith_number":"pith:2IUVRPNR","canonical_record":{"source":{"id":"1105.0178","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-01T15:29:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e8776d70d92e3650aa83fc619ddff38ee9164eca2f639903c7c22c2457912e59","abstract_canon_sha256":"63730816b774e768df6bd4b8d379fc1489dc4f3a2d633b901b9be987e5ebdf64"},"schema_version":"1.0"},"canonical_sha256":"d22958bdb1938de39c54bccae5a96a205274da8eda0b1a4c1f46d26fc7735981","source":{"kind":"arxiv","id":"1105.0178","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.0178","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"arxiv_version","alias_value":"1105.0178v1","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0178","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"pith_short_12","alias_value":"2IUVRPNRSOG6","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2IUVRPNRSOG6HHCU","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2IUVRPNR","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:2IUVRPNRSOG6HHCUXTFOLKLKEB","target":"record","payload":{"canonical_record":{"source":{"id":"1105.0178","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-01T15:29:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e8776d70d92e3650aa83fc619ddff38ee9164eca2f639903c7c22c2457912e59","abstract_canon_sha256":"63730816b774e768df6bd4b8d379fc1489dc4f3a2d633b901b9be987e5ebdf64"},"schema_version":"1.0"},"canonical_sha256":"d22958bdb1938de39c54bccae5a96a205274da8eda0b1a4c1f46d26fc7735981","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:12.376002Z","signature_b64":"yCwg5ClX8x/+90lK1qhqRgcDjf0Hqc0azMYsy90qjOfW1rPeFka2WJME+r1ZrELHF2ojRCTJA2vTcGvwW2wkAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d22958bdb1938de39c54bccae5a96a205274da8eda0b1a4c1f46d26fc7735981","last_reissued_at":"2026-05-18T04:23:12.375512Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:12.375512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1105.0178","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":"2026-05-18T04:23:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NaEBJ6WsWv1nrUokbwD/bwwZYI3jmRMS+XU5La7LiHAyFqyR02GZ/RYFdFPGU6ZCiMj6fux+EeAjdUxhcrfkAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T22:47:01.995052Z"},"content_sha256":"3be9cdade35c8acc5cb46436b2ed854164d093aac03a1a042663539ab236e043","schema_version":"1.0","event_id":"sha256:3be9cdade35c8acc5cb46436b2ed854164d093aac03a1a042663539ab236e043"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:2IUVRPNRSOG6HHCUXTFOLKLKEB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Abel-Zeilberger Algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Hai-Tao Jin, Qing-Hu Hou, William Y. C. Chen","submitted_at":"2011-05-01T15:29:55Z","abstract_excerpt":"We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0178","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":"2026-05-18T04:23:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IBbJ+bqS6/TgptSme7FFOrflg9rWgnLXBF9oq3JDO0WGAAk6YusKPq/0AL3sWhC5iCKJNmM4eBON/detcfgWBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T22:47:01.995740Z"},"content_sha256":"69f7f45ae2e0bb8bebc753c4018f899ee092cec4b1d9a841ee978163ca776ca5","schema_version":"1.0","event_id":"sha256:69f7f45ae2e0bb8bebc753c4018f899ee092cec4b1d9a841ee978163ca776ca5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2IUVRPNRSOG6HHCUXTFOLKLKEB/bundle.json","state_url":"https://pith.science/pith/2IUVRPNRSOG6HHCUXTFOLKLKEB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2IUVRPNRSOG6HHCUXTFOLKLKEB/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":"2026-06-07T22:47:01Z","links":{"resolver":"https://pith.science/pith/2IUVRPNRSOG6HHCUXTFOLKLKEB","bundle":"https://pith.science/pith/2IUVRPNRSOG6HHCUXTFOLKLKEB/bundle.json","state":"https://pith.science/pith/2IUVRPNRSOG6HHCUXTFOLKLKEB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2IUVRPNRSOG6HHCUXTFOLKLKEB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:2IUVRPNRSOG6HHCUXTFOLKLKEB","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":"63730816b774e768df6bd4b8d379fc1489dc4f3a2d633b901b9be987e5ebdf64","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-01T15:29:55Z","title_canon_sha256":"e8776d70d92e3650aa83fc619ddff38ee9164eca2f639903c7c22c2457912e59"},"schema_version":"1.0","source":{"id":"1105.0178","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.0178","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"arxiv_version","alias_value":"1105.0178v1","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0178","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"pith_short_12","alias_value":"2IUVRPNRSOG6","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2IUVRPNRSOG6HHCU","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2IUVRPNR","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:69f7f45ae2e0bb8bebc753c4018f899ee092cec4b1d9a841ee978163ca776ca5","target":"graph","created_at":"2026-05-18T04:23:12Z","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 use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.","authors_text":"Hai-Tao Jin, Qing-Hu Hou, William Y. C. Chen","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-01T15:29:55Z","title":"The Abel-Zeilberger Algorithm"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0178","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":"sha256:3be9cdade35c8acc5cb46436b2ed854164d093aac03a1a042663539ab236e043","target":"record","created_at":"2026-05-18T04:23:12Z","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":"63730816b774e768df6bd4b8d379fc1489dc4f3a2d633b901b9be987e5ebdf64","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-01T15:29:55Z","title_canon_sha256":"e8776d70d92e3650aa83fc619ddff38ee9164eca2f639903c7c22c2457912e59"},"schema_version":"1.0","source":{"id":"1105.0178","kind":"arxiv","version":1}},"canonical_sha256":"d22958bdb1938de39c54bccae5a96a205274da8eda0b1a4c1f46d26fc7735981","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d22958bdb1938de39c54bccae5a96a205274da8eda0b1a4c1f46d26fc7735981","first_computed_at":"2026-05-18T04:23:12.375512Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:12.375512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yCwg5ClX8x/+90lK1qhqRgcDjf0Hqc0azMYsy90qjOfW1rPeFka2WJME+r1ZrELHF2ojRCTJA2vTcGvwW2wkAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:12.376002Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.0178","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3be9cdade35c8acc5cb46436b2ed854164d093aac03a1a042663539ab236e043","sha256:69f7f45ae2e0bb8bebc753c4018f899ee092cec4b1d9a841ee978163ca776ca5"],"state_sha256":"7e10cc29432af0cb0f105588a634033a32d99b6355133f51f0936ae36546d6b3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SM5nCpQuKilPSS2iVTF6p2E6UOOO97W9/UAUuU3otQn1xbuWtVD/WurpzQCjykb4HaXb/NA4sz7ChiFJ80QqCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T22:47:01.999729Z","bundle_sha256":"9c9115284e605f52127837f9380ba026d87abef33ee9dd9b549954c0ba4c4a2b"}}