{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:21:49Z","timestamp":1760149309542,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T00:00:00Z","timestamp":1690416000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The paper deals with the problem of representation of Horn\u2019s hypergeometric functions via continued fractions and branched continued fractions. We construct the formal continued fraction expansions for three ratios of Horn\u2019s hypergeometric functions H7. The method employed is a two-dimensional generalization of the classical method of constructing a Gaussian continued fraction. It is proved that the continued fraction, which is an expansion of each ratio, uniformly converges to a holomorphic function of two variables on every compact subset of some domain of C2, and that this function is an analytic continuation of such a ratio in this domain. To illustrate this, we provide some numerical experiments at the end.<\/jats:p>","DOI":"10.3390\/axioms12080738","type":"journal-article","created":{"date-parts":[[2023,7,28]],"date-time":"2023-07-28T02:10:45Z","timestamp":1690510245000},"page":"738","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Representation of Some Ratios of Horn\u2019s Hypergeometric Functions H7 by Continued Fractions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0358-4641","authenticated-orcid":false,"given":"Tamara","family":"Antonova","sequence":"first","affiliation":[{"name":"Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Stepan Bandera Str., 79013 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2845-0137","authenticated-orcid":false,"given":"Roman","family":"Dmytryshyn","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Sciences, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-8517-4557","authenticated-orcid":false,"given":"Pavlo","family":"Kril","sequence":"additional","affiliation":[{"name":"Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Stepan Bandera Str., 79013 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2547-1442","authenticated-orcid":false,"given":"Serhii","family":"Sharyn","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Sciences, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,27]]},"reference":[{"key":"ref_1","first-page":"296","article-title":"Sur les s\u00e9ries hyper g\u00e9om\u00e9triques de deux variables et sur des \u00e9quations diff\u00e9rentielles lineaires aux d\u00e9riv\u00e9es partielles","volume":"90","author":"Appell","year":"1880","journal-title":"C. 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