{"paper":{"title":"Some basic bilateral sums and integrals","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Mizan Rahman, Mourad E. H. Ismail","submitted_at":"1993-11-01T00:00:00Z","abstract_excerpt":"By splitting the real line into intervals of unit length a doubly infinite integral of the form $\\Int F(q^x)\\,dx,\\; 0<q<1$, can clearly be expressed as $\\Integ \\Sum F(q^{x+n})\\,dx$, provided $F$ satisfies the appropriate conditions. This simple idea is used to prove Ramanujan's integral analogues of his \\ph{1}{1} sum and give a new proof of Askey and Roy's extention of it. Integral analogues of the well-poised \\ph{2}{2} sum as well as the very-well-poised \\ph{6}{6} sum are also found in a straightforward manner. An extension to a very-well-poised and balanced \\ph{8}{8} series is also given. A "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9311209","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"}