{"paper":{"title":"Explicit formulae for all higher order exponential lacunary generating functions of Hermite polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MP"],"primary_cat":"math-ph","authors_text":"G\\'erard H. E. Duchamp, Karol A. Penson, Nicolas Behr","submitted_at":"2018-06-21T20:22:20Z","abstract_excerpt":"For a sequence $P=(p_n(x))_{n=0}^{\\infty}$ of polynomials $p_n(x)$, we study the $K$-tuple and $L$-shifted exponential lacunary generating functions $\\mathcal{G}_{K,L}(\\lambda;x):=\\sum_{n=0}^{\\infty}\\frac{\\lambda^n}{n!} p_{n\\cdot K+L}(x)$, for $K=1,2\\dotsc$ and $L=0,1,2\\dotsc$. We establish an algorithm for efficiently computing $\\mathcal{G}_{K,L}(\\lambda;x)$ for generic polynomial sequences $P$. This procedure is exemplified by application to the study of Hermite polynomials, whereby we obtain closed-form expressions for $\\mathcal{G}_{K,L}(\\lambda;x)$ for arbitrary $K$ and $L$, in the form of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.08417","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"}