{"paper":{"title":"Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Juan F. Ma\\~nas-Ma\\~nas, Juan J. Moreno--Balc\\'azar, Lance L. Littlejohn, Richard Wellman","submitted_at":"2017-05-23T10:37:01Z","abstract_excerpt":"We consider the following discrete Sobolev inner product involving the Gegenbauer weight $$(f,g)_S:=\\int_{-1}^1f(x)g(x)(1-x^2)^{\\alpha}dx+M\\big[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)\\big],$$ where $\\alpha>-1,$ $j\\in \\mathbb{N}\\cup \\{0\\},$ and $M>0.$ Let $\\{Q_n^{(\\alpha,M,j)}\\}_{n\\geq0}$ be the sequence of orthogonal polynomials with respect to the above inner product. These polynomials are eigenfunctions of a differential operator $\\mathbf{T}. $ We establish the asymptotic behavior of the corresponding eigenvalues. Furthermore, we calculate the exact value $$r_0 = \\lim_{n\\rightarrow \\inft"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.08167","kind":"arxiv","version":1},"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"}