{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T19:43:31Z","timestamp":1769888611092,"version":"3.49.0"},"reference-count":24,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,20]],"date-time":"2023-04-20T00:00:00Z","timestamp":1681948800000},"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>Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we define the generalized (p,q)-Bernoulli\u2013Fibonacci and generalized (p,q)-Bernoulli\u2013Lucas polynomials and numbers by using the (p,q)-Bernoulli numbers, unified (p,q)-Bernoulli polynomials, h(x)-Fibonacci polynomials, and h(x)-Lucas polynomials. We also introduce the generalized bivariate (p,q)-Bernoulli\u2013Fibonacci and generalized bivariate (p,q)-Bernoulli\u2013Lucas polynomials and numbers. Then, we derive some properties of these newly established polynomials and numbers by using their generating functions with their functional equations. Finally, we provide some families of bilinear and bilateral generating functions for the generalized bivariate (p,q)-Bernoulli\u2013Fibonacci polynomials.<\/jats:p>","DOI":"10.3390\/sym15040943","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T03:01:48Z","timestamp":1682046108000},"page":"943","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["On Generalized Bivariate (p,q)-Bernoulli\u2013Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli\u2013Lucas Polynomials"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7859-5409","authenticated-orcid":false,"given":"Hao","family":"Guan","sequence":"first","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"},{"name":"School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Duyun 558000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem Ahmad","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7958-4226","authenticated-orcid":false,"given":"Can","family":"K\u0131z\u0131late\u015f","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zonguldak B\u00fclent Ecevit University, Zonguldak 67100, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Koshy, T. (2019). Fibonacci and Lucas Numbers with Applications, Wiley.","DOI":"10.1002\/9781118742297"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3179","DOI":"10.1016\/j.chaos.2009.04.048","article-title":"On generalized Fibonacci and Lucas polynomials","volume":"42","author":"Nalli","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1155\/2012\/264842","article-title":"Some properties of the (p,q)-Fibonacci and (p,q)-Lucas polynomials","volume":"2012","author":"Lee","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1890","DOI":"10.1016\/j.chaos.2007.09.071","article-title":"On the m-extension of the Fibonacci and Lucas p-numbers","volume":"40","author":"Kocer","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","first-page":"1243","article-title":"On some properties of Horadam polynomials","volume":"25","author":"Horzum","year":"2009","journal-title":"Int. Math. Forum."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1080\/00150517.1965.12431416","article-title":"Basic properties of a certain generalized sequence of numbers","volume":"3","author":"Horadam","year":"1965","journal-title":"Fibonacci Quart."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"14371","DOI":"10.1002\/mma.7702","article-title":"New families of Horadam numbers associated with finite operators and their applications","volume":"44","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"929","DOI":"10.55730\/1300-0098.3133","article-title":"On a class of generalized Humbert\u2013Hermite polynomials via generalized Fibonacci polynomials","volume":"46","author":"Pathan","year":"2022","journal-title":"Turk. J. Math."},{"key":"ref_9","first-page":"117","article-title":"On h(x)-Fibonacci-Euler and h(x)-Lucas-Euler numbers and polynomials","volume":"58","author":"Pathan","year":"2019","journal-title":"Acta Univ. Apulensis Math. Inform."},{"key":"ref_10","first-page":"8778","article-title":"Some properties of bi-variate bi-periodic Lucas polynomials","volume":"25","author":"Bala","year":"2021","journal-title":"Ann. Rom. Soc. Cell Biol."},{"key":"ref_11","unstructured":"Catalani, M. (2004). Generalized bivariate Fibonacci polynomials. arXiv."},{"key":"ref_12","first-page":"142","article-title":"Generalized Identities of Bivariate Fibonacci and Bivariate Lucas Polynomials","volume":"1","author":"Panwar","year":"2020","journal-title":"Jauist"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Alt\u0131nkaya, \u015e., Yal\u00e7\u0131n, S., and \u00c7akmak, S. (2019). A Subclass of Bi-Univalent Functions Based on the Faber Polynomial Expansions and the Fibonacci Numbers. Mathematics, 7.","DOI":"10.3390\/math7020160"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1007\/s40590-018-0212-z","article-title":"On the (p,q)-Lucas polynomial coefficient bounds of the bi-univalent function class \u03c3","volume":"25","year":"2019","journal-title":"Bol. Soc. Mat. Mex."},{"key":"ref_15","first-page":"79","article-title":"On \u03bb-Pseudo Bi-Starlike Functions related (p,q)-Lucas polynomial","volume":"39","author":"Murugusundaramoorthy","year":"2019","journal-title":"Lib. Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers.","DOI":"10.1016\/B978-0-12-385218-2.00002-5"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.jnt.2015.05.019","article-title":"On p-Bernoulli numbers and polynomials","volume":"157","author":"Rahmani","year":"2015","journal-title":"J. Number Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"975","DOI":"10.2989\/16073606.2017.1418762","article-title":"A closed formula for the generating function of p-Bernoulli numbers","volume":"41","author":"Rahmani","year":"2018","journal-title":"Quaest. Math."},{"key":"ref_19","first-page":"125","article-title":"Unified (p,q)-Bernoulli-Hermite polynomials","volume":"61","author":"Pathan","year":"2018","journal-title":"Fasc. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1080\/10652460500444928","article-title":"A unified presentation of some families of multivariable polynomials","volume":"17","author":"Erkus","year":"2006","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_21","unstructured":"Srivastava, H.M., and Manocha, H.L. (1984). A Treatise on Generating Functions, Wiley."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3253","DOI":"10.1007\/s13398-019-00687-4","article-title":"A parametric kind of the Fubini-type polynomials","volume":"113","author":"Srivastava","year":"2019","journal-title":"Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat. RACSAM"},{"key":"ref_23","first-page":"815","article-title":"Generating functions for the generalized bivariate Fibonacci and Lucas polynomials","volume":"18","author":"Tuglu","year":"2015","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"K\u0131z\u0131late\u015f, C. (2023). Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert-Hermite polynomials via generalized Fibonacci Polynomials. Math. Meth. Appl. Sci., 1\u201312.","DOI":"10.1002\/mma.9048"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/943\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:19:50Z","timestamp":1760123990000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/943"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,20]]},"references-count":24,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["sym15040943"],"URL":"https:\/\/doi.org\/10.3390\/sym15040943","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,20]]}}}