{"paper":{"title":"A kind of orthogonal polynomials and related identities II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Zhi-Hong Sun","submitted_at":"2017-11-16T08:44:43Z","abstract_excerpt":"For $n=0,1,2,\\ldots$ let $d_n^{(r)}(x)=\\sum_{k=0}^n\\binom{x+r+k}k\\binom{x-r}{n-k}$. In this paper we illustrate the connection between $\\{d_n^{(r)}(x)\\}$ and Meixner polynomials. New formulas and recurrence relations for $d_n^{(r)}(x)$ are obtained, and a new proof of the formula for $d_n^{(r)}(x)^2$ is also given. In addition, for $r>-\\frac 12$ and $n\\ge 2$ we show that $d_n^{(r)}(x)>\\frac{(2x+1)^n}{n!}>0$ for $x>-\\frac 12$, and $(-1)^nd_n^{(r)}(x)>0$ for $x<-\\frac 12$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05985","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"}