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Dar","submitted_at":"2026-05-11T08:10:36Z","abstract_excerpt":"In this paper, we obtain analytical evaluations of the Ramanujan integral \\[\\textbf{R}_{C}(m,n)= \\int_{0}^{\\infty}\\frac{x^m\\,\\cos(\\pi nx)}{\\exp{(2\\pi\\sqrt{x})-1}}dx\\] subject to suitable convergence conditions in terms of an infinite series of Meijer $G$-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function. %and Laplace transform method. We also consider some generalizations of the integral $\\textbf{R}_{C}(m,n)$ given as the integrals $I_{C}^{*}(\\upsilon,b,c,\\lambda,y)$ ,$\\Xi_{C}(\\upsilon,b,c,\\lambda,y)$, $\\nabla_{C}(\\upsilon,b,c,\\lambd"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.13882","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-11T08:10:36Z","cross_cats_sorted":[],"title_canon_sha256":"518c5e41b22b27f6b899201f7b03db2e077c2cd1135ed0977efb3ba069cb2475","abstract_canon_sha256":"034669c2e92015ef3a271268a185ae77845cb88678799e32677e895a01ab69f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:19.173326Z","signature_b64":"oQNq6oLvT/lpEUaLqHcFYF0j/W+eJ9ii3PToipWk7GjdgZuoPTFvvhotS6c5wSPNRIx3WFB19eL5QQ9tFiQxBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bf3924852a3637e622b453d7a635abffe79a5ee110df1bbbcf37f472e5fb8e5","last_reissued_at":"2026-05-17T23:39:19.172614Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:19.172614Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Generalized Closed Form of Ramanujan-Type Fourier Cosine Transform via Meijer's G-Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions under convergence conditions on m and n.","cross_cats":[],"primary_cat":"math.NT","authors_text":"R. 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We also consider some generalizations of the integral $\\textbf{R}_{C}(m,n)$ given as the integrals $I_{C}^{*}(\\upsilon,b,c,\\lambda,y)$ ,$\\Xi_{C}(\\upsilon,b,c,\\lambda,y)$, $\\nabla_{C}(\\upsilon,b,c,\\lambd"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We obtain analytical evaluations of the Ramanujan integral R_C(m,n) subject to suitable convergence conditions in terms of an infinite series of Meijer G-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The interchange of integration order and the absolute convergence of the resulting contour integrals under the stated conditions on m and n.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions, with generalizations and closed forms for nine related series.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions under convergence conditions on m and n.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b5ad7b59a663c7989c209bec4615a239b55769a02d0b912b8a160aa61cc6ea99"},"source":{"id":"2605.13882","kind":"arxiv","version":1},"verdict":{"id":"4e821ee8-7eb1-4d2b-b0ae-ba52d5d3fd13","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T05:57:50.950038Z","strongest_claim":"We obtain analytical evaluations of the Ramanujan integral R_C(m,n) subject to suitable convergence conditions in terms of an infinite series of Meijer G-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function.","one_line_summary":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions, with generalizations and closed forms for nine related series.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The interchange of integration order and the absolute convergence of the resulting contour integrals under the stated conditions on m and n.","pith_extraction_headline":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions under convergence conditions on m and n."},"references":{"count":20,"sample":[{"doi":"","year":2021,"title":"C., and Straub, A","work_id":"2a767044-e0a1-4252-8e43-4a014994e3a6","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Berndt, B. 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