{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T13:27:03Z","timestamp":1769002023867,"version":"3.49.0"},"reference-count":27,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T00:00:00Z","timestamp":1748476800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007414","name":"Deanship of Graduate Studies and Scientific Research at Qassim University","doi-asserted-by":"publisher","award":["QU-APC-2025"],"award-info":[{"award-number":["QU-APC-2025"]}],"id":[{"id":"10.13039\/501100007414","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This manuscript associates with a study of Frobenius\u2013Euler\u2013Simsek-type Polynomials. In this research work, we construct a new sequence of Sz\u00e1sz\u2013Beta type operators via Frobenius\u2013Euler\u2013Simsek-type Polynomials to discuss approximation properties for the Lebesgue integrable functions, i.e., Lp[0,\u221e), 1\u2264p&lt;\u221e. Furthermore, estimates in view of test functions and central moments are studied. Next, rate of convergence is discussed with the aid of the Korovkin theorem and the Voronovskaja type theorem. Moreover, direct approximation results in terms of modulus of continuity of first- and second-order, Peetre\u2019s K-functional, Lipschitz type space, and the rth-order Lipschitz type maximal functions are investigated. In the subsequent section, we present weighted approximation results, and statistical approximation theorems are discussed. To demonstrate the effectiveness and applicability of the proposed operators, we present several illustrative examples and visualize the results graphically.<\/jats:p>","DOI":"10.3390\/axioms14060418","type":"journal-article","created":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T06:25:04Z","timestamp":1748499904000},"page":"418","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Sz\u00e1sz\u2013Beta Operators Linking Frobenius\u2013Euler\u2013Simsek-Type Polynomials"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5681-9563","authenticated-orcid":false,"given":"Nadeem","family":"Rao","sequence":"first","affiliation":[{"name":"Department of Mathematics, University Center for Research and Development, Chandigarh University, Mohali 140413, Punjab, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-3509","authenticated-orcid":false,"given":"Mohammad","family":"Farid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7805-4362","authenticated-orcid":false,"given":"Shivani","family":"Bansal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University Institute of Sciences, Chandigarh University, Mohali 140413, Punjab, India"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,29]]},"reference":[{"key":"ref_1","unstructured":"Weierstrass, K. 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