{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:Q3UDUCX6Z5HOBMC4AXNW4XDP5R","short_pith_number":"pith:Q3UDUCX6","canonical_record":{"source":{"id":"math/0609283","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-09-11T10:37:42Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"05e22b6a01156ab539d4d1575314eaeea04cd4271d02803bb164ccc4e171c80c","abstract_canon_sha256":"42f48a7c587daf180221ec552b3a63a8c8d2a020ce82df779ed5f31bba324156"},"schema_version":"1.0"},"canonical_sha256":"86e83a0afecf4ee0b05c05db6e5c6fec4ce80b611c983dc8c071bd81f627d760","source":{"kind":"arxiv","id":"math/0609283","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609283","created_at":"2026-05-18T04:42:34Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609283v2","created_at":"2026-05-18T04:42:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609283","created_at":"2026-05-18T04:42:34Z"},{"alias_kind":"pith_short_12","alias_value":"Q3UDUCX6Z5HO","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"Q3UDUCX6Z5HOBMC4","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"Q3UDUCX6","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:Q3UDUCX6Z5HOBMC4AXNW4XDP5R","target":"record","payload":{"canonical_record":{"source":{"id":"math/0609283","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-09-11T10:37:42Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"05e22b6a01156ab539d4d1575314eaeea04cd4271d02803bb164ccc4e171c80c","abstract_canon_sha256":"42f48a7c587daf180221ec552b3a63a8c8d2a020ce82df779ed5f31bba324156"},"schema_version":"1.0"},"canonical_sha256":"86e83a0afecf4ee0b05c05db6e5c6fec4ce80b611c983dc8c071bd81f627d760","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:34.669073Z","signature_b64":"prv58UXz9+aHvPAcnkYPiuaU04MT0nnribGgk6h5MfSL13On/yj5GRPHgL3Q+8Mj3FrHsw34LLXKBjIC4WUKBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86e83a0afecf4ee0b05c05db6e5c6fec4ce80b611c983dc8c071bd81f627d760","last_reissued_at":"2026-05-18T04:42:34.668423Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:34.668423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0609283","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:42:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+oDVYsPF9RYUFc7DIssuzC7fWW5BdaDYhxGcy/HX/i/+qirS6cSuHls6fJOD3GVL7+mLyJkMG/m55+5KYZsJDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T06:04:18.919553Z"},"content_sha256":"0aba4d8f961a671212ff0b656952e92d28628c23deaff325f1dc0f97be29fd92","schema_version":"1.0","event_id":"sha256:0aba4d8f961a671212ff0b656952e92d28628c23deaff325f1dc0f97be29fd92"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:Q3UDUCX6Z5HOBMC4AXNW4XDP5R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fibonacci numbers and orthogonal polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Christian Berg","submitted_at":"2006-09-11T10:37:42Z","abstract_excerpt":"We prove that the sequence $(1/F_{n+2})_{n\\ge 0}$ of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little $q$-Jacobi polynomials with $q=(1-\\sqrt{5})/(1+\\sqrt{5})$. We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices $(1/F_{i+j+2})$ have integer entries. We prove analogous results for the Hilbert matrices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609283","kind":"arxiv","version":2},"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":"2026-05-18T04:42:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2E/Ok0FOISszBc6HeuegylqoRa0cIiYkf8sVISdbmnyVIAhbrlF4N2GZVl+VBeWfmouisnZgCK1PDeDQqGqSDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T06:04:18.919899Z"},"content_sha256":"a1a52b34c2f981f6bbd95f168909beaa9fc740d7ba6e1d7c3b7a1a3abde509eb","schema_version":"1.0","event_id":"sha256:a1a52b34c2f981f6bbd95f168909beaa9fc740d7ba6e1d7c3b7a1a3abde509eb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q3UDUCX6Z5HOBMC4AXNW4XDP5R/bundle.json","state_url":"https://pith.science/pith/Q3UDUCX6Z5HOBMC4AXNW4XDP5R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q3UDUCX6Z5HOBMC4AXNW4XDP5R/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":"2026-06-21T06:04:18Z","links":{"resolver":"https://pith.science/pith/Q3UDUCX6Z5HOBMC4AXNW4XDP5R","bundle":"https://pith.science/pith/Q3UDUCX6Z5HOBMC4AXNW4XDP5R/bundle.json","state":"https://pith.science/pith/Q3UDUCX6Z5HOBMC4AXNW4XDP5R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q3UDUCX6Z5HOBMC4AXNW4XDP5R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:Q3UDUCX6Z5HOBMC4AXNW4XDP5R","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":"42f48a7c587daf180221ec552b3a63a8c8d2a020ce82df779ed5f31bba324156","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-09-11T10:37:42Z","title_canon_sha256":"05e22b6a01156ab539d4d1575314eaeea04cd4271d02803bb164ccc4e171c80c"},"schema_version":"1.0","source":{"id":"math/0609283","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609283","created_at":"2026-05-18T04:42:34Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609283v2","created_at":"2026-05-18T04:42:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609283","created_at":"2026-05-18T04:42:34Z"},{"alias_kind":"pith_short_12","alias_value":"Q3UDUCX6Z5HO","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"Q3UDUCX6Z5HOBMC4","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"Q3UDUCX6","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:a1a52b34c2f981f6bbd95f168909beaa9fc740d7ba6e1d7c3b7a1a3abde509eb","target":"graph","created_at":"2026-05-18T04:42:34Z","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 prove that the sequence $(1/F_{n+2})_{n\\ge 0}$ of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little $q$-Jacobi polynomials with $q=(1-\\sqrt{5})/(1+\\sqrt{5})$. We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices $(1/F_{i+j+2})$ have integer entries. We prove analogous results for the Hilbert matrices.","authors_text":"Christian Berg","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-09-11T10:37:42Z","title":"Fibonacci numbers and orthogonal polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609283","kind":"arxiv","version":2},"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":"sha256:0aba4d8f961a671212ff0b656952e92d28628c23deaff325f1dc0f97be29fd92","target":"record","created_at":"2026-05-18T04:42:34Z","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":"42f48a7c587daf180221ec552b3a63a8c8d2a020ce82df779ed5f31bba324156","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2006-09-11T10:37:42Z","title_canon_sha256":"05e22b6a01156ab539d4d1575314eaeea04cd4271d02803bb164ccc4e171c80c"},"schema_version":"1.0","source":{"id":"math/0609283","kind":"arxiv","version":2}},"canonical_sha256":"86e83a0afecf4ee0b05c05db6e5c6fec4ce80b611c983dc8c071bd81f627d760","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"86e83a0afecf4ee0b05c05db6e5c6fec4ce80b611c983dc8c071bd81f627d760","first_computed_at":"2026-05-18T04:42:34.668423Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:34.668423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"prv58UXz9+aHvPAcnkYPiuaU04MT0nnribGgk6h5MfSL13On/yj5GRPHgL3Q+8Mj3FrHsw34LLXKBjIC4WUKBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:34.669073Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0609283","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0aba4d8f961a671212ff0b656952e92d28628c23deaff325f1dc0f97be29fd92","sha256:a1a52b34c2f981f6bbd95f168909beaa9fc740d7ba6e1d7c3b7a1a3abde509eb"],"state_sha256":"3f3cbee682a6597ec5f1db1886428a806434a264c31f5e6c33d51d99c9694329"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HcCQ7S7LTLIsf/zJBHFIKftiFsHpnP6mGDZrmx13xmg4PX2e4iD35343iyYtrgugTy3bgMJW5o6lOwM2uetyBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T06:04:18.921814Z","bundle_sha256":"bdb2b1e7c7033f70ceffc3c3545e273b11657a4d5e9529e6498b7c5c40d89d43"}}