{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1994:LDNBDPOAWNLV6EE2UR4RNKQWAQ","short_pith_number":"pith:LDNBDPOA","canonical_record":{"source":{"id":"math/9412227","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"1994-12-17T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"697f4e5cdd91575b09add824e5d9017dbed403c3cddd75ef9dba6c0bee77c6a1","abstract_canon_sha256":"099d18f17208c4134f3c8f38a203fdb1a6dba8aef11e63a37d782d8736a196db"},"schema_version":"1.0"},"canonical_sha256":"58da11bdc0b3575f109aa47916aa16041803de72341d41ca90cc9e4a30b82959","source":{"kind":"arxiv","id":"math/9412227","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9412227","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/9412227v1","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9412227","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"pith_short_12","alias_value":"LDNBDPOAWNLV","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"LDNBDPOAWNLV6EE2","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"LDNBDPOA","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1994:LDNBDPOAWNLV6EE2UR4RNKQWAQ","target":"record","payload":{"canonical_record":{"source":{"id":"math/9412227","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"1994-12-17T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"697f4e5cdd91575b09add824e5d9017dbed403c3cddd75ef9dba6c0bee77c6a1","abstract_canon_sha256":"099d18f17208c4134f3c8f38a203fdb1a6dba8aef11e63a37d782d8736a196db"},"schema_version":"1.0"},"canonical_sha256":"58da11bdc0b3575f109aa47916aa16041803de72341d41ca90cc9e4a30b82959","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:50.716380Z","signature_b64":"Qu3hHp5uTv5HRslr/B1MevTKin6wqj6GspA5Wclnl8+rscR5QIpVgD0EYVWhxLf0Rio+J1843A/wTgoqoY91BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58da11bdc0b3575f109aa47916aa16041803de72341d41ca90cc9e4a30b82959","last_reissued_at":"2026-05-18T01:05:50.715870Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:50.715870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9412227","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":"2026-05-18T01:05:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FcymBCFCJXrgE0SliPlJEAIDMLQ++4HSfTibLUwIOcg78r6rlHrBff6gZ+Z2O0P1Y5IqUigShk1uYTPEaJ7qCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T23:41:34.843934Z"},"content_sha256":"8027f236556e5eccc73696ad83310313699afe9120bad23b17727422cc3e64b5","schema_version":"1.0","event_id":"sha256:8027f236556e5eccc73696ad83310313699afe9120bad23b17727422cc3e64b5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1994:LDNBDPOAWNLV6EE2UR4RNKQWAQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algorithms for the indefinite and definite summation","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Wolfram Koepf","submitted_at":"1994-12-17T00:00:00Z","abstract_excerpt":"The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both $F(n+1,k)/F(n,k)$ and $F(n,k+1)/F(n,k)$ are rational functions. Typical examples are ratios of products of exponentials, factorials, $\\Gamma$ function terms, bin omial coefficients, and Pochhammer symbols that are integer-linear with respect to $n$ and $k$ in their arguments.\n  We consider the more general case of ratios of products of exponentials, factori"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9412227","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":"2026-05-18T01:05:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j0kbTwDK+7RDXi1PJiQq/gthUkJAJ7TvtQ/bc0/U6a9asSrL+sruBhcEMcw2glvhqa1hbtCGWCWqTc3PMecDBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T23:41:34.844545Z"},"content_sha256":"0404f3354bf0687999c0109d8f2ff3c9b1951de1b6d7a59e1c230b42fabdd29b","schema_version":"1.0","event_id":"sha256:0404f3354bf0687999c0109d8f2ff3c9b1951de1b6d7a59e1c230b42fabdd29b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LDNBDPOAWNLV6EE2UR4RNKQWAQ/bundle.json","state_url":"https://pith.science/pith/LDNBDPOAWNLV6EE2UR4RNKQWAQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LDNBDPOAWNLV6EE2UR4RNKQWAQ/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":"2026-07-01T23:41:34Z","links":{"resolver":"https://pith.science/pith/LDNBDPOAWNLV6EE2UR4RNKQWAQ","bundle":"https://pith.science/pith/LDNBDPOAWNLV6EE2UR4RNKQWAQ/bundle.json","state":"https://pith.science/pith/LDNBDPOAWNLV6EE2UR4RNKQWAQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LDNBDPOAWNLV6EE2UR4RNKQWAQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:LDNBDPOAWNLV6EE2UR4RNKQWAQ","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":"099d18f17208c4134f3c8f38a203fdb1a6dba8aef11e63a37d782d8736a196db","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"1994-12-17T00:00:00Z","title_canon_sha256":"697f4e5cdd91575b09add824e5d9017dbed403c3cddd75ef9dba6c0bee77c6a1"},"schema_version":"1.0","source":{"id":"math/9412227","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9412227","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/9412227v1","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9412227","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"pith_short_12","alias_value":"LDNBDPOAWNLV","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"LDNBDPOAWNLV6EE2","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"LDNBDPOA","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:0404f3354bf0687999c0109d8f2ff3c9b1951de1b6d7a59e1c230b42fabdd29b","target":"graph","created_at":"2026-05-18T01:05:50Z","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":"The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both $F(n+1,k)/F(n,k)$ and $F(n,k+1)/F(n,k)$ are rational functions. Typical examples are ratios of products of exponentials, factorials, $\\Gamma$ function terms, bin omial coefficients, and Pochhammer symbols that are integer-linear with respect to $n$ and $k$ in their arguments.\n  We consider the more general case of ratios of products of exponentials, factori","authors_text":"Wolfram Koepf","cross_cats":[],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"1994-12-17T00:00:00Z","title":"Algorithms for the indefinite and definite summation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9412227","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":"sha256:8027f236556e5eccc73696ad83310313699afe9120bad23b17727422cc3e64b5","target":"record","created_at":"2026-05-18T01:05:50Z","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":"099d18f17208c4134f3c8f38a203fdb1a6dba8aef11e63a37d782d8736a196db","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"1994-12-17T00:00:00Z","title_canon_sha256":"697f4e5cdd91575b09add824e5d9017dbed403c3cddd75ef9dba6c0bee77c6a1"},"schema_version":"1.0","source":{"id":"math/9412227","kind":"arxiv","version":1}},"canonical_sha256":"58da11bdc0b3575f109aa47916aa16041803de72341d41ca90cc9e4a30b82959","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58da11bdc0b3575f109aa47916aa16041803de72341d41ca90cc9e4a30b82959","first_computed_at":"2026-05-18T01:05:50.715870Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:50.715870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qu3hHp5uTv5HRslr/B1MevTKin6wqj6GspA5Wclnl8+rscR5QIpVgD0EYVWhxLf0Rio+J1843A/wTgoqoY91BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:50.716380Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9412227","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8027f236556e5eccc73696ad83310313699afe9120bad23b17727422cc3e64b5","sha256:0404f3354bf0687999c0109d8f2ff3c9b1951de1b6d7a59e1c230b42fabdd29b"],"state_sha256":"aa8cba2bf38722fe525e9831f877779005981d231c5bc81ab62ad48350fccad5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AXf4dp/kbSm9nIed6LiSd+6bvY0jHbS/1CJ3MQZGr3CKFhfyI0D/SFqV9FiUhSIK32Fy57UWEVQksMfO+F6NCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T23:41:34.847744Z","bundle_sha256":"818d6e754ab734289de0b0f22510acbe757e456753f0b1d4f9dead4be0c2b051"}}