{"paper":{"title":"A kind of orthogonal polynomials and related identities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2016-06-27T15:48:38Z","abstract_excerpt":"In this paper we introduce the polynomials $\\{d_n^{(r)}(x)\\}$ and $\\{D_n^{(r)}(x)\\}$ given by $d_n^{(r)}(x)=\\sum_{k=0}^n\\binom{x+r+k}k\\binom{x-r}{n-k} \\ (n\\ge 0)$, $D_0^{(r)}(x)=1,\\ D_1^{(r)}(x)=x$ and $D_{n+1}^{(r)}(x)=xD_n^{(r)}(x)-n(n+2r)D_{n-1}^{(r)}(x)\\ (n\\ge 1).$ We show that $\\{D_n^{(r)}(x)\\}$ are orthogonal polynomials for $r>-\\frac 12$, and establish many identities for $\\{d_n^{(r)}(x)\\}$ and $\\{D_n^{(r)}(x)\\}$, especially obtain a formula for $d_n^{(r)}(x)^2$ and the linearization formulas for $d_m^{(r)}(x)d_n^{(r)}(x)$ and $D_m^{(r)}(x)D_n^{(r)}(x)$. As an application we extend rece"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.08327","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"}