{"paper":{"title":"The Parabolic Mellin Transform: Gamma and Zeta Integral Representations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.PR"],"primary_cat":"math.NT","authors_text":"Chen Tong, Peter Reinhard Hansen","submitted_at":"2026-02-19T02:09:32Z","abstract_excerpt":"We introduce the Parabolic Mellin Transform (PMT), defined by ${P}_{\\sigma}[f](z)=\\int_{-\\infty}^{\\infty}w^{2z}f(w^2)dt$, where $w=\\sigma+it$ and $\\sigma>0$. Under the substitution $u=w^2$, the vertical line $\\operatorname{Re}(w)=\\sigma$ is mapped to the parabolic contour $C_\\sigma$ in the $u$-plane. For the Gaussian kernel, the PMT yields $\\int_{-\\infty}^{\\infty}w^{2z}e^{w^2}dt=\\pi/\\Gamma(\\tfrac{1}{2}-z)=\\cos(\\pi z)\\Gamma(z+\\tfrac{1}{2})$, a parabolic-contour form of the classical Hankel representation for the reciprocal Gamma function. The advantage of this parametrization is that the contou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.17007","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.17007/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}