{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T00:50:33Z","timestamp":1776905433028,"version":"3.51.2"},"reference-count":28,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T00:00:00Z","timestamp":1732492800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Graduate Studies and Scientific Research at Qassim University","award":["QU-APC-2024-9\/1"],"award-info":[{"award-number":["QU-APC-2024-9\/1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In 1812, Gauss stated the following identity: F12(a,b;c;1)=\u0393(c)\u0393(c\u2212a\u2212b)\u0393(c\u2212a)\u0393(c\u2212b), where, in the real case, c\u2212a\u2212b&gt;0 and as an immediat consequence the Chu\u2013Vandermonde identity: F12(a,\u2212n;c;1)=(c\u2212a)n(c)n for any positive integer n. In this paper, we investigate the case when c\u2212a\u2212b&lt;0 by taking c=2b=\u22122n, n and a are positive integers (c\u2212a\u2212b=\u2212n\u2212a&lt;0). We give two significant applications stemming from these findings. The second part of the paper will be devoted to Kummer\u2019s conditions concerning hypergeometric quadratic transformations, particularly focusing on the distinctions between the conditions provided by Gradshteyn and Ryzhik (GR) and those by Erd\u00e9lyi, Magnus, Oberhettinger, and Tricomi (EMOI) are outlined. We establish that the conditions given by GR differ from those of EMOI, and we explore the methodologies employed by both groups in deriving their results. This leads us to conclude that the search for exact and unified conditions remains an open problem.<\/jats:p>","DOI":"10.3390\/axioms13120825","type":"journal-article","created":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T11:25:33Z","timestamp":1732533933000},"page":"825","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Extension of Chu\u2013Vandermonde Identity and Quadratic Transformation Conditions"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3331-6063","authenticated-orcid":false,"given":"Mohamed Jalel","family":"Atia","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-7430-4009","authenticated-orcid":false,"given":"Maged","family":"Alkilayh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,25]]},"reference":[{"key":"ref_1","unstructured":"Gauss, C.F. 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Special Functions, Macmillan."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Atia, M.J. (2022). Resolution of an Isolated Case of a Quadratic Hypergeometric 2F1 Transformation. Axioms, 11.","DOI":"10.3390\/axioms11100533"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Atia, M.J., and Al-Mohaimeed, A.S. (2023). On a resolution of another isolated case of a Kummer\u2019s quadratic transformation for 2F1. Axioms, 12.","DOI":"10.3390\/axioms12020221"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Atia, M.J., and Rathie, A.K. (2023). On a Generalization of the Kummer\u2019s Quadratic Transformation and a Resolution of an Isolated Case. Axioms, 12.","DOI":"10.3390\/axioms12090821"},{"key":"ref_26","unstructured":"Slater, L.J. (1966). Generalized Hypergeometric Functions, Cambridge University Press."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Magnus, W., Oberhettinger, F., and Soni, R.P. (1966). 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DLMF: NIST Digital Library of Mathematical Functions, Available online: https:\/\/dlmf.nist.gov\/."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/825\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:39:18Z","timestamp":1760114358000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/825"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,25]]},"references-count":28,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["axioms13120825"],"URL":"https:\/\/doi.org\/10.3390\/axioms13120825","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,25]]}}}