{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T15:46:24Z","timestamp":1781711184626,"version":"3.54.5"},"reference-count":25,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,30]],"date-time":"2022-05-30T00:00:00Z","timestamp":1653868800000},"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>Special polynomials play an important role in several subjects of mathematics, engineering, and theoretical physics. Many problems arising in mathematics, engineering, and mathematical physics are framed in terms of differential equations. In this paper, we introduce the family of the Lagrange-based hypergeometric Bernoulli polynomials via the generating function method. We state some algebraic and differential properties for this family of extensions of the Lagrange-based Bernoulli polynomials, as well as a matrix-inversion formula involving these polynomials. Moreover, a generating relation involving the Stirling numbers of the second kind was derived. In fact, future investigations in this subject could be addressed for the potential applications of these polynomials in the aforementioned disciplines.<\/jats:p>","DOI":"10.3390\/sym14061125","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T02:30:06Z","timestamp":1653964206000},"page":"1125","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Lagrange-Based Hypergeometric Bernoulli Polynomials"],"prefix":"10.3390","volume":"14","author":[{"given":"Sahar","family":"Albosaily","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1053-0892","authenticated-orcid":false,"given":"Yamilet","family":"Quintana","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Legan\u00e9s, 28911 Madrid, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5103-6092","authenticated-orcid":false,"given":"Azhar","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem A.","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"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1080\/10652460108819340","article-title":"The Lagrange polynomials in several variables","volume":"12","author":"Chan","year":"2001","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_2","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. (1953). Higher Transcendental Functions, McGraw Hill."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Mathai, A.M., and Haubold, H.J. (2008). Special Functions for Applied Scientists, Springer Science Business Media.","DOI":"10.1007\/978-0-387-75894-7"},{"key":"ref_4","unstructured":"Szeg\u00f6, G. (1939). Orthogonal Polynomials, American Math. Soc."},{"key":"ref_5","unstructured":"Srivastava, H.M., and Manocha, H.L. (1984). A Treatise on Generating Functions, Ellis Horwood Ltd."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"415","DOI":"10.11650\/twjm\/1500407662","article-title":"Multidimensional extensions of the Bernoulli and Appell polynomials","volume":"8","author":"Bretti","year":"2004","journal-title":"Taiwanese J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1155\/S1085337504306263","article-title":"Generalizations of the Bernoulli and Appell polynomials","volume":"7","author":"Bretti","year":"2004","journal-title":"Abstr. Appl. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1155\/S1110757X03204101","article-title":"A generalization of the Bernoulli polynomials","volume":"2003","author":"Natalini","year":"2003","journal-title":"J. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1134\/S106192081301010X","article-title":"Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials","volume":"10","author":"Srivastava","year":"2013","journal-title":"Russian J. Math. Phys."},{"key":"ref_10","first-page":"1","article-title":"Generalized hypergeometric Bernoulli numbers","volume":"115","author":"Chakraborty","year":"2021","journal-title":"Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Math. RACSAM"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1142\/S1793042108001754","article-title":"Hypergeometric Bernoulli polynomials and Appell sequences","volume":"4","author":"Hassen","year":"2008","journal-title":"Int. J. Number Theory"},{"key":"ref_12","first-page":"701","article-title":"Some sequences of rational numbers related to the exponential function","volume":"34","author":"Howard","year":"1967","journal-title":"Duke Math. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10092-018-0272-5","article-title":"On an operational matrix method based on generalized Bernoulli polynomials of level m","volume":"55","author":"Quintana","year":"2018","journal-title":"Calcolo"},{"key":"ref_14","first-page":"1","article-title":"Some new classes of generalized Lagrange-based Apostol type Hermite polynomials","volume":"10","author":"Khan","year":"2019","journal-title":"J. Inequal. Spec. Funct."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"865","DOI":"10.46793\/KgJMat2206.865K","article-title":"On generalized Lagrange-based Apostol-type and related polynomials","volume":"46","author":"Khan","year":"2022","journal-title":"Kragujevac J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"227","DOI":"10.18576\/amis\/120122","article-title":"The Lagrange polynomials in several variables","volume":"12","author":"Duran","year":"2018","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"245","DOI":"10.46793\/KgJMat2302.245Q","article-title":"Generalized mixed type Bernoulli-Gegenbauer polynomial","volume":"47","author":"Quintana","year":"2020","journal-title":"Kragujevac J. Math."},{"key":"ref_18","first-page":"1","article-title":"A new class of degenerate Apostol-type Hermite polynomials and applications","volume":"15","author":"Cesarano","year":"2022","journal-title":"Dolomites Res. Notes Approx."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/978-3-030-69236-0_6","article-title":"A note on Hermite-Bernoulli Polynomials","volume":"Volume 26","author":"Beghin","year":"2021","journal-title":"Nonlocal and Fractional Operators"},{"key":"ref_20","first-page":"111","article-title":"Laguerre-based Hermite-Bernoulli polynomials associated with bilateral series","volume":"11","author":"Khan","year":"2018","journal-title":"Tbilisi Math. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s00025-014-0430-2","article-title":"About extensions of generalized Apostol-type polynomials","volume":"68","author":"Quintana","year":"2015","journal-title":"Results Math."},{"key":"ref_22","first-page":"157","article-title":"Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials","volume":"55","author":"Navas","year":"2019","journal-title":"Arch. Math."},{"key":"ref_23","first-page":"1","article-title":"Special values of Lerch zeta function and their Fourier expansions","volume":"21","author":"Bayad","year":"2011","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2679","DOI":"10.1002\/mma.6075","article-title":"On the multidimensional zeta functions associated with theta functions, and the multidimensional Appell polynomials","volume":"43","author":"Bayad","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1093\/qmathj\/haaa006","article-title":"On a formula for the regularized determinant of zeta functions with application to some Dirichlet series","volume":"71","author":"Hajli","year":"2020","journal-title":"Q. J. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1125\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:21:06Z","timestamp":1760138466000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1125"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,30]]},"references-count":25,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061125"],"URL":"https:\/\/doi.org\/10.3390\/sym14061125","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,30]]}}}