{"paper":{"title":"An integral identity with applications in orthogonal polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Yuan Xu","submitted_at":"2014-05-12T15:39:21Z","abstract_excerpt":"For $\\boldsymbol{\\large {\\lambda}} = (\\lambda_1,\\ldots,\\lambda_d)$ with $\\lambda_i > 0$, it is proved that \\begin{equation*}\n  \\prod_{i=1}^d \\frac{ 1}{(1- r x_i)^{\\lambda_i}} = \\frac{\\Gamma(|\\boldsymbol{\\large {\\lambda}}|)}{\\prod_{i=1}^{d} \\Gamma(\\lambda_i)} \\int_{\\mathcal{T}^d} \\frac{1}{ (1- r \\langle x, u \\rangle)^{|\\boldsymbol{\\large {\\lambda}}|}} \\prod_{i=1}^d u_i^{\\lambda_i-1} du, \\end{equation*} where $\\mathcal{T}^d$ is the simplex in homogeneous coordinates of $\\mathbb{R}^d$, from which a new integral relation for Gegenbuer polynomials of different indexes is deduced. The latter result "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2812","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"}