{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T00:50:22Z","timestamp":1774918222457,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T00:00:00Z","timestamp":1659484800000},"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>The main purpose of this paper is to define a new family of Sz\u00e1sz\u2013Mirakyan operators that depends on a non-negative parameter, say \u03b1. This new family of Sz\u00e1sz\u2013Mirakyan operators is crucial in that it includes both the existing Sz\u00e1sz\u2013Mirakyan operator and allows the construction of new operators for different values of \u03b1. Then, the convergence properties of the new operators with the aid of the Popoviciu\u2013Bohman\u2013Korovkin theorem-type property are presented. The Voronovskaja-type theorem and rate of convergence are provided in a detailed proof. Furthermore, with the help of the classical modulus of continuity, we deduce an upper bound for the error of the new operator. In addition to these, in order to show that the convex or monotonic functions produced convex or monotonic operators, we obtain shape-preserving properties of the new family of Sz\u00e1sz\u2013Mirakyan operators. The symmetry of the properties of the classical Sz\u00e1sz\u2013Mirakyan operator and of the properties of the new sequence is investigated. Moreover, we compare this operator with its classical correspondence to show that the new one has superior properties. Finally, some numerical illustrative examples are presented to strengthen our theoretical results.<\/jats:p>","DOI":"10.3390\/sym14081596","type":"journal-article","created":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T23:33:01Z","timestamp":1659569581000},"page":"1596","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["A Generalization of Sz\u00e1sz\u2013Mirakyan Operators Based on \u03b1 Non-Negative Parameter"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4564-6211","authenticated-orcid":false,"given":"Khursheed J.","family":"Ansari","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7750-6910","authenticated-orcid":false,"given":"Fuat","family":"Usta","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, D\u00fczce University, D\u00fczce 81620, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,3]]},"reference":[{"key":"ref_1","first-page":"633","article-title":"\u00dcber die analytische Darstellbarkeit sogenannter willk\u00fcrlicher funktionen einer reellen Veranderlichen","volume":"2","author":"Weierstrass","year":"1885","journal-title":"Sitzungsberichte Akad. 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