{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T13:27:11Z","timestamp":1774618031706,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T00:00:00Z","timestamp":1678838400000},"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 by branched continued fractions. The formal branched continued fraction expansions for three different Horn\u2019s hypergeometric function H4 ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain H,H\u2282C2, and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn\u2019s double hypergeometric series H4 is considered, and the expression of solutions of some systems of partial differential equations is indicated.<\/jats:p>","DOI":"10.3390\/axioms12030299","type":"journal-article","created":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T05:22:59Z","timestamp":1678857779000},"page":"299","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["On Some Branched Continued Fraction Expansions for Horn\u2019s Hypergeometric Function H4(a,b;c,d;z1,z2) Ratios"],"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\/0000-0002-4100-2972","authenticated-orcid":false,"given":"Ilona-Anna","family":"Lutsiv","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\/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":[[2023,3,15]]},"reference":[{"key":"ref_1","unstructured":"Bodnarchuk, P.I., and Skorobogatko, V.Y. 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