{"paper":{"title":"Irrationality proof of a $q$-extension of $\\zeta(2)$ using little $q$-Jacobi polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Christophe Smet, Walter Van Assche","submitted_at":"2008-09-15T12:14:36Z","abstract_excerpt":"We show how one can use Hermite-Pad\\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\\zeta_q(2)$. These numbers are $q$-analogues of the well known $\\zeta(2)$. Here $q=\\frac{1}{p}$, with $p$ an integer greater than one. These approximants are good enough to show the irrationality of $\\zeta_q(2)$ and they allow us to calculate an upper bound for its measure of irrationality: $\\mu(\\zeta_q(2))\\leq 10\\pi^2/(5\\pi^2-24) \\approx 3.8936$. This is sharper than the upper bound given by Zudilin (\\textit{On the irrationality measure for a $q$-analogue of $\\zeta("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.2501","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"}