{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:40:43Z","timestamp":1760031643494,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,31]],"date-time":"2025-03-31T00:00:00Z","timestamp":1743379200000},"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 this paper, we introduce and study new hybrid generalizations of bivariate-balancing and Lucas-balancing polynomials. We give some properties of bivariate-balancing and Lucas-balancing hybrinomials, among other Binet-type formulas, Catalan, Cassini, d\u2019Ocagne, Vajda, Ruggles, and Honsberger identities. Moreover, we present matrix generators and generating functions of these hybrinomials.<\/jats:p>","DOI":"10.3390\/sym17040537","type":"journal-article","created":{"date-parts":[[2025,4,2]],"date-time":"2025-04-02T05:43:33Z","timestamp":1743572613000},"page":"537","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Bivariate-Balancing and Lucas-Balancing Hybrinomials"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4554-4875","authenticated-orcid":false,"given":"Mariola","family":"Rubajczyk","sequence":"first","affiliation":[{"name":"The Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, Al. Powsta\u0144c\u00f3w Warszawy 12, 35-959 Rzesz\u00f3w, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5508-0640","authenticated-orcid":false,"given":"Anetta","family":"Szynal-Liana","sequence":"additional","affiliation":[{"name":"The Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, Al. Powsta\u0144c\u00f3w Warszawy 12, 35-959 Rzesz\u00f3w, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1080\/00150517.1999.12428864","article-title":"On the square roots of triangular numbers","volume":"37","author":"Behera","year":"1999","journal-title":"Fibonacci Quart."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Br\u00f3d, D., Rubajczyk, M., and Szynal-Liana, A. (2024). A new hybrid generalization of balancing polynomials. 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