{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:NPQNWMVZIBSE6WJMVV5F2AHLRT","short_pith_number":"pith:NPQNWMVZ","canonical_record":{"source":{"id":"1112.6111","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-28T12:53:24Z","cross_cats_sorted":[],"title_canon_sha256":"ee113ee06de33c87aaebd4672d077f1b52776d7f1622e13908819c73f79f8d74","abstract_canon_sha256":"608ac3047fc60d7599241055fb8378a4e92854e6ca935a3609c261c5931db6fb"},"schema_version":"1.0"},"canonical_sha256":"6be0db32b940644f592cad7a5d00eb8cfc84bab793ed259d0854227d97ab306e","source":{"kind":"arxiv","id":"1112.6111","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.6111","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"arxiv_version","alias_value":"1112.6111v1","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.6111","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"pith_short_12","alias_value":"NPQNWMVZIBSE","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NPQNWMVZIBSE6WJM","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NPQNWMVZ","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:NPQNWMVZIBSE6WJMVV5F2AHLRT","target":"record","payload":{"canonical_record":{"source":{"id":"1112.6111","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-28T12:53:24Z","cross_cats_sorted":[],"title_canon_sha256":"ee113ee06de33c87aaebd4672d077f1b52776d7f1622e13908819c73f79f8d74","abstract_canon_sha256":"608ac3047fc60d7599241055fb8378a4e92854e6ca935a3609c261c5931db6fb"},"schema_version":"1.0"},"canonical_sha256":"6be0db32b940644f592cad7a5d00eb8cfc84bab793ed259d0854227d97ab306e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:36.760823Z","signature_b64":"gJxGC+h3oELJV1/jh1NuGiI/rmZrF6Z9zHP8N4rpomDt1vqWpjOx3k7cE1QFmYIOO1CljF2mQgw5gMXxCYKBCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6be0db32b940644f592cad7a5d00eb8cfc84bab793ed259d0854227d97ab306e","last_reissued_at":"2026-05-18T04:05:36.760181Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:36.760181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.6111","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":"2026-05-18T04:05:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TALDCkRcRqrM74golNZpv2EHjh86nnQPlxdSeEcwfh2utewQwOZNGnUb8g2vkkXFjFBFWMTC8fd7ds0Bq0jKCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T13:41:23.875974Z"},"content_sha256":"d197ba4807bcd52250f3cd34c7c81fff00ed9dff8345ff9f5737ccb103bb8e68","schema_version":"1.0","event_id":"sha256:d197ba4807bcd52250f3cd34c7c81fff00ed9dff8345ff9f5737ccb103bb8e68"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:NPQNWMVZIBSE6WJMVV5F2AHLRT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Jacobi-Stirling Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eric S. Egge, George E. Andrews, Lance L. Littlejohn, Wolfgang Gawronski","submitted_at":"2011-12-28T12:53:24Z","abstract_excerpt":"The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which, as shown in LW, are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its compan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.6111","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":"2026-05-18T04:05:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EuO7+V9uIBfvM+sC1078AhslcpT+M1s70gCGg/TaFlo7VMQsmpKuw3Vc0QgXw2l4s0lW+WyJU9T8pbi0tb/SDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T13:41:23.876306Z"},"content_sha256":"8816e1b5c5724d7e4c6e315a89fb6f202ece84bfa0e57cb8f2c79f4ec019a704","schema_version":"1.0","event_id":"sha256:8816e1b5c5724d7e4c6e315a89fb6f202ece84bfa0e57cb8f2c79f4ec019a704"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NPQNWMVZIBSE6WJMVV5F2AHLRT/bundle.json","state_url":"https://pith.science/pith/NPQNWMVZIBSE6WJMVV5F2AHLRT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NPQNWMVZIBSE6WJMVV5F2AHLRT/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":"2026-06-25T13:41:23Z","links":{"resolver":"https://pith.science/pith/NPQNWMVZIBSE6WJMVV5F2AHLRT","bundle":"https://pith.science/pith/NPQNWMVZIBSE6WJMVV5F2AHLRT/bundle.json","state":"https://pith.science/pith/NPQNWMVZIBSE6WJMVV5F2AHLRT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NPQNWMVZIBSE6WJMVV5F2AHLRT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:NPQNWMVZIBSE6WJMVV5F2AHLRT","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":"608ac3047fc60d7599241055fb8378a4e92854e6ca935a3609c261c5931db6fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-28T12:53:24Z","title_canon_sha256":"ee113ee06de33c87aaebd4672d077f1b52776d7f1622e13908819c73f79f8d74"},"schema_version":"1.0","source":{"id":"1112.6111","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.6111","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"arxiv_version","alias_value":"1112.6111v1","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.6111","created_at":"2026-05-18T04:05:36Z"},{"alias_kind":"pith_short_12","alias_value":"NPQNWMVZIBSE","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NPQNWMVZIBSE6WJM","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NPQNWMVZ","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:8816e1b5c5724d7e4c6e315a89fb6f202ece84bfa0e57cb8f2c79f4ec019a704","target":"graph","created_at":"2026-05-18T04:05:36Z","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 Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which, as shown in LW, are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its compan","authors_text":"Eric S. Egge, George E. Andrews, Lance L. Littlejohn, Wolfgang Gawronski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-28T12:53:24Z","title":"The Jacobi-Stirling Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.6111","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:d197ba4807bcd52250f3cd34c7c81fff00ed9dff8345ff9f5737ccb103bb8e68","target":"record","created_at":"2026-05-18T04:05:36Z","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":"608ac3047fc60d7599241055fb8378a4e92854e6ca935a3609c261c5931db6fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-28T12:53:24Z","title_canon_sha256":"ee113ee06de33c87aaebd4672d077f1b52776d7f1622e13908819c73f79f8d74"},"schema_version":"1.0","source":{"id":"1112.6111","kind":"arxiv","version":1}},"canonical_sha256":"6be0db32b940644f592cad7a5d00eb8cfc84bab793ed259d0854227d97ab306e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6be0db32b940644f592cad7a5d00eb8cfc84bab793ed259d0854227d97ab306e","first_computed_at":"2026-05-18T04:05:36.760181Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:36.760181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gJxGC+h3oELJV1/jh1NuGiI/rmZrF6Z9zHP8N4rpomDt1vqWpjOx3k7cE1QFmYIOO1CljF2mQgw5gMXxCYKBCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:36.760823Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.6111","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d197ba4807bcd52250f3cd34c7c81fff00ed9dff8345ff9f5737ccb103bb8e68","sha256:8816e1b5c5724d7e4c6e315a89fb6f202ece84bfa0e57cb8f2c79f4ec019a704"],"state_sha256":"e73f3d117a64270e2ba1b5e44bf8626c3b850f6e24f330b58f03ddafddd6a78c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QR0scJdKdU1+oCHwZSoRkM0OPoT15LAt2XkBD9Lp2fC3HzUbpeS1aWMXHpSbUJLD6zsBNMKxAG7IeaM+urVKCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T13:41:23.878140Z","bundle_sha256":"49653705f3f9279d2baf61fa6acc8d670ee9a287ebf6ffde584d3d86a8e32476"}}