{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T11:01:00Z","timestamp":1770980460308,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,13]],"date-time":"2022-01-13T00:00:00Z","timestamp":1642032000000},"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 the present analysis, we aim to construct a new subclass of analytic bi-univalent functions defined on symmetric domain by means of the Pascal distribution series and Gegenbauer polynomials. Thereafter, we provide estimates of Taylor\u2013Maclaurin coefficients a2 and a3 for functions in the aforementioned class, and next, we solve the Fekete\u2013Szeg\u00f6 functional problem. Moreover, some interesting findings for new subclasses of analytic bi-univalent functions will emerge by reducing the parameters in our main results.<\/jats:p>","DOI":"10.3390\/sym14010147","type":"journal-article","created":{"date-parts":[[2022,1,14]],"date-time":"2022-01-14T03:14:56Z","timestamp":1642130096000},"page":"147","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":36,"title":["Exploiting the Pascal Distribution Series and Gegenbauer Polynomials to Construct and Study a New Subclass of Analytic Bi-Univalent Functions"],"prefix":"10.3390","volume":"14","author":[{"given":"Ala","family":"Amourah","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8608-8063","authenticated-orcid":false,"given":"Basem Aref","family":"Frasin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al Al-Bayt University, Mafraq 25113, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Morad","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Jordan, Amman 11942, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2740-7081","authenticated-orcid":false,"given":"Feras","family":"Yousef","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Jordan, Amman 11942, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bain, L., and Engelhardt, M. (1992). Introduction to Probability and Mathematical Statistics, Duxburry Press.","DOI":"10.2307\/2532587"},{"key":"ref_2","first-page":"301","article-title":"Pascal Distribution Series Connected with Certain Subclasses of Univalent Functions","volume":"59","author":"Bulboaca","year":"2019","journal-title":"Kyungpook Math. J."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Agarwal, P., Agarwal, R.P., and Ruzhansky, M. (2020). Special Functions and Analysis of Differential Equations, CRC Press.","DOI":"10.1201\/9780429320026"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ruzhansky, M., Cho, Y.J., Agarwal, P., and Area, I. (2017). Advances in Real and Complex Analysis with Applications, Springer.","DOI":"10.1007\/978-981-10-4337-6"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Doman, B. (2015). The Classical Orthogonal Polynomials, World Scientific.","DOI":"10.1142\/9700"},{"key":"ref_6","unstructured":"Chihara, T.S. (2011). An Introduction to Orthogonal Polynomials, Courier Corporation."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ismail, M., Ismail, M.E., and van Assche, W. (2005). Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press.","DOI":"10.1017\/CBO9781107325982"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"403","DOI":"10.34198\/ejms.7221.403427","article-title":"New Families of Bi-univalent Functions Governed by Gegenbauer Polynomials","volume":"7","author":"Wanas","year":"2021","journal-title":"Earthline J. Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/s40590-019-00245-3","article-title":"A Comprehensive Subclass of Bi-univalent Functions Associated with Chebyshev Polynomials of the Second Kind","volume":"26","author":"Yousef","year":"2020","journal-title":"Bol. Soc. Mat. 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