{"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"}