{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,20]],"date-time":"2026-10-20T21:42:02Z","timestamp":1792532522837,"version":"4.5.0"},"reference-count":40,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,11]],"date-time":"2024-02-11T00:00:00Z","timestamp":1707609600000},"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>In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella\u2013Saran\u2019s hypergeometric function\u00a0FK\u00a0with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks.<\/jats:p>","DOI":"10.3390\/sym16020220","type":"journal-article","created":{"date-parts":[[2024,2,12]],"date-time":"2024-02-12T04:47:45Z","timestamp":1707713265000},"page":"220","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["On the Analytic Extension of Lauricella\u2013Saran\u2019s Hypergeometric Function FK to Symmetric Domains"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2845-0137","authenticated-orcid":false,"given":"Roman","family":"Dmytryshyn","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, Ukraine"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-3513-7304","authenticated-orcid":false,"given":"Vitaliy","family":"Goran","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, Ukraine"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF03012437","article-title":"Sulle funzioni ipergeometriche a pi\u00f9 variabili","volume":"7","author":"Lauricella","year":"1893","journal-title":"Rend. 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