{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:48:29Z","timestamp":1760230109956,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,9]],"date-time":"2022-07-09T00:00:00Z","timestamp":1657324800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The symmetric patterns that inequalities contain are reflected in researchers\u2019 studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function \u0393\u03bc(v) and study the completely monotonic (CM) property of a function involving \u0393\u03bc(v) and the generalized digamma function \u03c8\u03bc(v). As a consequence, we establish some bounds for \u0393\u03bc(v), \u03c8\u03bc(v) and polygamma functions \u03c8\u03bc(r)(v), r\u22651.<\/jats:p>","DOI":"10.3390\/sym14071412","type":"journal-article","created":{"date-parts":[[2022,7,11]],"date-time":"2022-07-11T00:06:21Z","timestamp":1657497981000},"page":"1412","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Asymptotic Expansion for the Generalized Gamma Function"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5918-1913","authenticated-orcid":false,"given":"Mansour","family":"Mahmoud","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hanan","family":"Almuashi","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hesham","family":"Moustafa","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,9]]},"reference":[{"key":"ref_1","unstructured":"Abramowitz, M., and Stegun, I.A. (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc."},{"key":"ref_2","unstructured":"Cichon, J. (2022, June 10). Stiriling Approximation Formula. 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Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1412\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:47:18Z","timestamp":1760140038000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1412"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,9]]},"references-count":22,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["sym14071412"],"URL":"https:\/\/doi.org\/10.3390\/sym14071412","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,7,9]]}}}