{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:B75LPE6NCG3TEE5UZPPNPOBVCJ","short_pith_number":"pith:B75LPE6N","canonical_record":{"source":{"id":"1303.4846","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-20T06:24:46Z","cross_cats_sorted":[],"title_canon_sha256":"a7c4473c10b81e2003f980dc94b7d12b5d6dfd8ff32e921d2de4d59653595c21","abstract_canon_sha256":"757baebd2bf1aac67024bc77e9a3ee848ccf9147ee136572834054bf171d8c64"},"schema_version":"1.0"},"canonical_sha256":"0ffab793cd11b73213b4cbded7b83512618bc0e59f841d7750e4087ecb1b7a51","source":{"kind":"arxiv","id":"1303.4846","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.4846","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"arxiv_version","alias_value":"1303.4846v3","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.4846","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"pith_short_12","alias_value":"B75LPE6NCG3T","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"B75LPE6NCG3TEE5U","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"B75LPE6N","created_at":"2026-05-18T12:27:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:B75LPE6NCG3TEE5UZPPNPOBVCJ","target":"record","payload":{"canonical_record":{"source":{"id":"1303.4846","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-20T06:24:46Z","cross_cats_sorted":[],"title_canon_sha256":"a7c4473c10b81e2003f980dc94b7d12b5d6dfd8ff32e921d2de4d59653595c21","abstract_canon_sha256":"757baebd2bf1aac67024bc77e9a3ee848ccf9147ee136572834054bf171d8c64"},"schema_version":"1.0"},"canonical_sha256":"0ffab793cd11b73213b4cbded7b83512618bc0e59f841d7750e4087ecb1b7a51","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:39.667774Z","signature_b64":"EbnVNB3KvMiEcKYKuNFN58lMEGrMqM3gf4m7HBp6QC5QMacV7n7QQtW2mRp4GCqpadPrUG3obfgMNABipkooCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ffab793cd11b73213b4cbded7b83512618bc0e59f841d7750e4087ecb1b7a51","last_reissued_at":"2026-05-18T02:54:39.667024Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:39.667024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.4846","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:54:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K8mEyFT2b5zt/Uwe+y2vArBuHzOS1gPruHlSF7HIOkc3oSar5BXX5rVtkM0veWp5UhzNpQcGtnexXeWtONl1BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T05:36:43.679700Z"},"content_sha256":"f3b16ed0cb1cb3cfbb7a52f9d348fd66402b4549f47ebfcd7386ea1b4ac2f4de","schema_version":"1.0","event_id":"sha256:f3b16ed0cb1cb3cfbb7a52f9d348fd66402b4549f47ebfcd7386ea1b4ac2f4de"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:B75LPE6NCG3TEE5UZPPNPOBVCJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linear Difference Equations with a Transition Point at the Origin","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Lihua Cao, Yutian Li","submitted_at":"2013-03-20T06:24:46Z","abstract_excerpt":"A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation {equation*} P_{n+1}(x)-(A_{n}x+B_{n})P_{n}(x)+P_{n-1}(x)=0, {equation*} where $A_n$ and $B_n$ have asymptotic expansions of the form {equation*} A_n\\sim n^{-\\theta}\\sum_{s=0}^\\infty\\frac{\\alpha_s}{n^s},\\qquad B_n\\sim\\sum_{s=0}^\\infty\\frac{\\beta_s}{n^s}, {equation*} with $\\theta\\neq0$ and $\\alpha_0\\neq0$ being real numbers, and $\\beta_0=\\pm2$. Our result hold uniformly for the scaled variable $t$ in an infinite interval containing the transition point $t_1=0$, where $t=(n+\\tau_0)^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4846","kind":"arxiv","version":3},"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":"2026-05-18T02:54:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CWoqxGCyIV7UakyLCjxqTyfMou7bDUclrdCST2aXmbzXSVZAXI6KpyoS/hnUn9A3fGigxc7bnhsbj18f/gIPDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T05:36:43.680102Z"},"content_sha256":"2737ea77dc4f7cbadc3652034f62201952843b83bb94b03695ca20f84a819600","schema_version":"1.0","event_id":"sha256:2737ea77dc4f7cbadc3652034f62201952843b83bb94b03695ca20f84a819600"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B75LPE6NCG3TEE5UZPPNPOBVCJ/bundle.json","state_url":"https://pith.science/pith/B75LPE6NCG3TEE5UZPPNPOBVCJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B75LPE6NCG3TEE5UZPPNPOBVCJ/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":"2026-06-07T05:36:43Z","links":{"resolver":"https://pith.science/pith/B75LPE6NCG3TEE5UZPPNPOBVCJ","bundle":"https://pith.science/pith/B75LPE6NCG3TEE5UZPPNPOBVCJ/bundle.json","state":"https://pith.science/pith/B75LPE6NCG3TEE5UZPPNPOBVCJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B75LPE6NCG3TEE5UZPPNPOBVCJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:B75LPE6NCG3TEE5UZPPNPOBVCJ","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":"757baebd2bf1aac67024bc77e9a3ee848ccf9147ee136572834054bf171d8c64","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-20T06:24:46Z","title_canon_sha256":"a7c4473c10b81e2003f980dc94b7d12b5d6dfd8ff32e921d2de4d59653595c21"},"schema_version":"1.0","source":{"id":"1303.4846","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.4846","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"arxiv_version","alias_value":"1303.4846v3","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.4846","created_at":"2026-05-18T02:54:39Z"},{"alias_kind":"pith_short_12","alias_value":"B75LPE6NCG3T","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"B75LPE6NCG3TEE5U","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"B75LPE6N","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:2737ea77dc4f7cbadc3652034f62201952843b83bb94b03695ca20f84a819600","target":"graph","created_at":"2026-05-18T02:54:39Z","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":"A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation {equation*} P_{n+1}(x)-(A_{n}x+B_{n})P_{n}(x)+P_{n-1}(x)=0, {equation*} where $A_n$ and $B_n$ have asymptotic expansions of the form {equation*} A_n\\sim n^{-\\theta}\\sum_{s=0}^\\infty\\frac{\\alpha_s}{n^s},\\qquad B_n\\sim\\sum_{s=0}^\\infty\\frac{\\beta_s}{n^s}, {equation*} with $\\theta\\neq0$ and $\\alpha_0\\neq0$ being real numbers, and $\\beta_0=\\pm2$. Our result hold uniformly for the scaled variable $t$ in an infinite interval containing the transition point $t_1=0$, where $t=(n+\\tau_0)^","authors_text":"Lihua Cao, Yutian Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-20T06:24:46Z","title":"Linear Difference Equations with a Transition Point at the Origin"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4846","kind":"arxiv","version":3},"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:f3b16ed0cb1cb3cfbb7a52f9d348fd66402b4549f47ebfcd7386ea1b4ac2f4de","target":"record","created_at":"2026-05-18T02:54:39Z","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":"757baebd2bf1aac67024bc77e9a3ee848ccf9147ee136572834054bf171d8c64","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-20T06:24:46Z","title_canon_sha256":"a7c4473c10b81e2003f980dc94b7d12b5d6dfd8ff32e921d2de4d59653595c21"},"schema_version":"1.0","source":{"id":"1303.4846","kind":"arxiv","version":3}},"canonical_sha256":"0ffab793cd11b73213b4cbded7b83512618bc0e59f841d7750e4087ecb1b7a51","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0ffab793cd11b73213b4cbded7b83512618bc0e59f841d7750e4087ecb1b7a51","first_computed_at":"2026-05-18T02:54:39.667024Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:39.667024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EbnVNB3KvMiEcKYKuNFN58lMEGrMqM3gf4m7HBp6QC5QMacV7n7QQtW2mRp4GCqpadPrUG3obfgMNABipkooCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:39.667774Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.4846","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f3b16ed0cb1cb3cfbb7a52f9d348fd66402b4549f47ebfcd7386ea1b4ac2f4de","sha256:2737ea77dc4f7cbadc3652034f62201952843b83bb94b03695ca20f84a819600"],"state_sha256":"a3643c78220bfde5f3e159555084a5d7bd52e1f19250daac2288abe4f14ce6f4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zldTidI1U81pMbZwsNrws61FFIyDfPQtg61htQnfmNJ/rdSsZ2v1khpEYaL9L+eVBwyXbKHEeKTzPP50pfvlCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T05:36:43.683487Z","bundle_sha256":"b63b8b94e8a312c35777124c008d2fde62c089ff06ba550852e8c3a394f0b8a5"}}