{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:XQQTRF7O3XPMHYTRHNZKW64KTY","short_pith_number":"pith:XQQTRF7O","schema_version":"1.0","canonical_sha256":"bc213897eedddec3e2713b72ab7b8a9e0c5c23ce399f510f4a99859b5ff40637","source":{"kind":"arxiv","id":"1701.00787","version":2},"attestation_state":"computed","paper":{"title":"Positive definite functions on the unit sphere and integrals of Jacobi polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Yuan Xu","submitted_at":"2017-01-03T19:03:02Z","abstract_excerpt":"It is shown that the integrals of the Jacobi polynomials \\begin{equation*}%\\label{eq:Fn^J}\n  \\int_0^t (t-\\theta)^\\delta P_n^{(\\alpha-\\frac12,\\beta-\\frac12)}(\\cos \\theta) \\left(\\sin \\tfrac{\\theta}2\\right)^{2 \\alpha}\n  \\left(\\cos \\tfrac{\\theta}2\\right)^{2 \\beta} d\\theta > 0 \\end{equation*} for all $t \\in (0,\\pi]$ and $n \\in \\mathbb{N}$ if $\\delta \\ge \\alpha + 1$ for $\\alpha,\\beta \\in \\mathbb{N}_0$ and $\\max\\{\\alpha,\\beta\\} > 0$. This proves a conjecture on the integral of the Gegenbauer polynomials in \\cite{BCX} that implies the strictly positive definiteness of the function $\\theta \\mapsto (t -"},"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":"1701.00787","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-01-03T19:03:02Z","cross_cats_sorted":[],"title_canon_sha256":"5175b26100ecb8df6c7b0ee66deeb776716d067bbf78200c6ae04d7e5267b5b5","abstract_canon_sha256":"31518b8406086d1a961178fe398b288a554fed1617cc734ffa94fbf9e8a62f1f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:41.849496Z","signature_b64":"SUjNi8xfDkmdTaqWHegzgShENXOAf2dMV7TAD7XpPODnUhrzLsWgKDmm8ORrFAl1fG17pMzev9DqNvOjPrzZCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc213897eedddec3e2713b72ab7b8a9e0c5c23ce399f510f4a99859b5ff40637","last_reissued_at":"2026-05-18T00:38:41.848828Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:41.848828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Positive definite functions on the unit sphere and integrals of Jacobi polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Yuan Xu","submitted_at":"2017-01-03T19:03:02Z","abstract_excerpt":"It is shown that the integrals of the Jacobi polynomials \\begin{equation*}%\\label{eq:Fn^J}\n  \\int_0^t (t-\\theta)^\\delta P_n^{(\\alpha-\\frac12,\\beta-\\frac12)}(\\cos \\theta) \\left(\\sin \\tfrac{\\theta}2\\right)^{2 \\alpha}\n  \\left(\\cos \\tfrac{\\theta}2\\right)^{2 \\beta} d\\theta > 0 \\end{equation*} for all $t \\in (0,\\pi]$ and $n \\in \\mathbb{N}$ if $\\delta \\ge \\alpha + 1$ for $\\alpha,\\beta \\in \\mathbb{N}_0$ and $\\max\\{\\alpha,\\beta\\} > 0$. This proves a conjecture on the integral of the Gegenbauer polynomials in \\cite{BCX} that implies the strictly positive definiteness of the function $\\theta \\mapsto (t -"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00787","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1701.00787","created_at":"2026-05-18T00:38:41.848917+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.00787v2","created_at":"2026-05-18T00:38:41.848917+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.00787","created_at":"2026-05-18T00:38:41.848917+00:00"},{"alias_kind":"pith_short_12","alias_value":"XQQTRF7O3XPM","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_16","alias_value":"XQQTRF7O3XPMHYTR","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_8","alias_value":"XQQTRF7O","created_at":"2026-05-18T12:31:56.362134+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/XQQTRF7O3XPMHYTRHNZKW64KTY","json":"https://pith.science/pith/XQQTRF7O3XPMHYTRHNZKW64KTY.json","graph_json":"https://pith.science/api/pith-number/XQQTRF7O3XPMHYTRHNZKW64KTY/graph.json","events_json":"https://pith.science/api/pith-number/XQQTRF7O3XPMHYTRHNZKW64KTY/events.json","paper":"https://pith.science/paper/XQQTRF7O"},"agent_actions":{"view_html":"https://pith.science/pith/XQQTRF7O3XPMHYTRHNZKW64KTY","download_json":"https://pith.science/pith/XQQTRF7O3XPMHYTRHNZKW64KTY.json","view_paper":"https://pith.science/paper/XQQTRF7O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.00787&json=true","fetch_graph":"https://pith.science/api/pith-number/XQQTRF7O3XPMHYTRHNZKW64KTY/graph.json","fetch_events":"https://pith.science/api/pith-number/XQQTRF7O3XPMHYTRHNZKW64KTY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XQQTRF7O3XPMHYTRHNZKW64KTY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XQQTRF7O3XPMHYTRHNZKW64KTY/action/storage_attestation","attest_author":"https://pith.science/pith/XQQTRF7O3XPMHYTRHNZKW64KTY/action/author_attestation","sign_citation":"https://pith.science/pith/XQQTRF7O3XPMHYTRHNZKW64KTY/action/citation_signature","submit_replication":"https://pith.science/pith/XQQTRF7O3XPMHYTRHNZKW64KTY/action/replication_record"}},"created_at":"2026-05-18T00:38:41.848917+00:00","updated_at":"2026-05-18T00:38:41.848917+00:00"}