{"paper":{"title":"Spectral analysis of two doubly infinite Jacobi matrices with exponential entries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Franti\\v{s}ek \\v{S}tampach, Mourad E. H. Ismail","submitted_at":"2016-05-02T08:49:11Z","abstract_excerpt":"We provide a complete spectral analysis of all self-adjoint operators acting on $\\ell^{2}(\\mathbb{Z})$ which are associated with two doubly infinite Jacobi matrices with entries given by $$ q^{-n+1}\\delta_{m,n-1}+q^{-n}\\delta_{m,n+1} $$ and $$ \\delta_{m,n-1}+\\alpha q^{-n}\\delta_{m,n}+\\delta_{m,n+1}, $$ respectively, where $q\\in(0,1)$ and $\\alpha\\in\\mathbb{R}$. As an application, we derive orthogonality relations for the Ramanujan entire function and the third Jackson $q$-Bessel function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.00393","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"}