{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:XI2BPDX7LCLFYUOVHPSYIEGCYG","short_pith_number":"pith:XI2BPDX7","schema_version":"1.0","canonical_sha256":"ba34178eff58965c51d53be58410c2c1a40e20a1fa38800f72c14f068589a106","source":{"kind":"arxiv","id":"math/0403344","version":4},"attestation_state":"computed","paper":{"title":"Chebyshev Series Expansion of Inverse Polynomials","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Richard J. Mathar","submitted_at":"2004-03-22T02:40:14Z","abstract_excerpt":"An inverse polynomial has a Chebyshev series expansion\n 1/\\sum(j=0..k)b_j*T_j(x)=\\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear combinations of simple functions of the polynomial roots. If the first k of the coefficients a_n are known, the others become linear combinations of these with expansion coefficients derived recursively from the b_j's. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the b_j in f"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0403344","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2004-03-22T02:40:14Z","cross_cats_sorted":[],"title_canon_sha256":"3054b620e01db7073c859a04165ef6c1730271db8063300c301ea50f626bff3a","abstract_canon_sha256":"2eacfa93c885c2e20251243cfd28aa58ce246e01318074b4246bbfc65ceb7bb2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:26.504752Z","signature_b64":"NHi0MxKurtgopy6FEg+ImnDFjtbhmFmuCpz9VTYEIgzwjPZvwDYb8hr7OulaGAIQaQYoiERQ/FpKd0NSm4LuBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba34178eff58965c51d53be58410c2c1a40e20a1fa38800f72c14f068589a106","last_reissued_at":"2026-05-18T01:05:26.504146Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:26.504146Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Chebyshev Series Expansion of Inverse Polynomials","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Richard J. Mathar","submitted_at":"2004-03-22T02:40:14Z","abstract_excerpt":"An inverse polynomial has a Chebyshev series expansion\n 1/\\sum(j=0..k)b_j*T_j(x)=\\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear combinations of simple functions of the polynomial roots. If the first k of the coefficients a_n are known, the others become linear combinations of these with expansion coefficients derived recursively from the b_j's. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the b_j in f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0403344","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0403344","created_at":"2026-05-18T01:05:26.504232+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0403344v4","created_at":"2026-05-18T01:05:26.504232+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0403344","created_at":"2026-05-18T01:05:26.504232+00:00"},{"alias_kind":"pith_short_12","alias_value":"XI2BPDX7LCLF","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_16","alias_value":"XI2BPDX7LCLFYUOV","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_8","alias_value":"XI2BPDX7","created_at":"2026-05-18T12:25:52.687210+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XI2BPDX7LCLFYUOVHPSYIEGCYG","json":"https://pith.science/pith/XI2BPDX7LCLFYUOVHPSYIEGCYG.json","graph_json":"https://pith.science/api/pith-number/XI2BPDX7LCLFYUOVHPSYIEGCYG/graph.json","events_json":"https://pith.science/api/pith-number/XI2BPDX7LCLFYUOVHPSYIEGCYG/events.json","paper":"https://pith.science/paper/XI2BPDX7"},"agent_actions":{"view_html":"https://pith.science/pith/XI2BPDX7LCLFYUOVHPSYIEGCYG","download_json":"https://pith.science/pith/XI2BPDX7LCLFYUOVHPSYIEGCYG.json","view_paper":"https://pith.science/paper/XI2BPDX7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0403344&json=true","fetch_graph":"https://pith.science/api/pith-number/XI2BPDX7LCLFYUOVHPSYIEGCYG/graph.json","fetch_events":"https://pith.science/api/pith-number/XI2BPDX7LCLFYUOVHPSYIEGCYG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XI2BPDX7LCLFYUOVHPSYIEGCYG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XI2BPDX7LCLFYUOVHPSYIEGCYG/action/storage_attestation","attest_author":"https://pith.science/pith/XI2BPDX7LCLFYUOVHPSYIEGCYG/action/author_attestation","sign_citation":"https://pith.science/pith/XI2BPDX7LCLFYUOVHPSYIEGCYG/action/citation_signature","submit_replication":"https://pith.science/pith/XI2BPDX7LCLFYUOVHPSYIEGCYG/action/replication_record"}},"created_at":"2026-05-18T01:05:26.504232+00:00","updated_at":"2026-05-18T01:05:26.504232+00:00"}