{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T01:12:04Z","timestamp":1767575524006,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,12]],"date-time":"2019-06-12T00:00:00Z","timestamp":1560297600000},"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, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell\u2013Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants.<\/jats:p>","DOI":"10.3390\/sym11060782","type":"journal-article","created":{"date-parts":[[2019,6,12]],"date-time":"2019-06-12T10:55:19Z","timestamp":1560336919000},"page":"782","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6239-2968","authenticated-orcid":false,"given":"Feng","family":"Qi","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, China"},{"name":"College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China"},{"name":"School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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, Faculty of Arts and Sciences, Zonguldak B\u00fclent Ecevit University, Zonguldak 67100, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8996-2270","authenticated-orcid":false,"given":"Wei-Shih","family":"Du","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,12]]},"reference":[{"key":"ref_1","first-page":"161","article-title":"Basic properties of a certain generalized sequence of numbers","volume":"3","author":"Horadam","year":"1965","journal-title":"Fibonacci Quart."},{"key":"ref_2","first-page":"137","article-title":"Jacobsthal representation polynomials","volume":"35","author":"Horadam","year":"1997","journal-title":"Fibonacci Quart."},{"key":"ref_3","first-page":"1243","article-title":"On some properties of Horadam polynomials","volume":"4","author":"Horzum","year":"2009","journal-title":"Int. Math. Forum"},{"key":"ref_4","unstructured":"Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications, Wiley-Interscience. Pure and Applied Mathematics."},{"key":"ref_5","unstructured":"Koshy, T. (2018). Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, Inc.. [2nd ed.]. Pure and Applied Mathematics."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1080\/07468342.2002.11921945","article-title":"Fibonacci determinants","volume":"33","author":"Cahill","year":"2002","journal-title":"Coll. Math. J."},{"key":"ref_7","first-page":"216","article-title":"Fibonacci and Lucas numbers as tridiagonal matrix determinants","volume":"42","author":"Cahill","year":"2004","journal-title":"Fibonacci Quart."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.laa.2015.10.004","article-title":"Inverse spectral problems for collections of leading principal submatrices of tridiagonal matrices","volume":"489","author":"Higgins","year":"2016","journal-title":"Linear Algebra Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/BF02161358","article-title":"Handbook Series Linear Algebra: Similarity reduction of a general matrix to Hessenberg form","volume":"12","author":"Martin","year":"1968","journal-title":"Numer. Math."},{"key":"ref_10","first-page":"287","article-title":"Alternative proofs of some formulas for two tridiagonal determinants","volume":"10","author":"Qi","year":"2018","journal-title":"Acta Univ. Sapientiae Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"17","DOI":"10.14419\/gjma.v4i3.6574","article-title":"A determinantal representation for derangement numbers","volume":"4","author":"Qi","year":"2016","journal-title":"Glob. J. Math. Anal."},{"key":"ref_12","unstructured":"Qi, F. (2020). Determinantal expressions and recurrence relations for Fubini and Eulerian polynomials. J. Interdiscip. Math., 22."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"844","DOI":"10.1016\/j.amc.2015.06.123","article-title":"Derivatives of tangent function and tangent numbers","volume":"268","author":"Qi","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_14","first-page":"123","article-title":"Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials","volume":"81","author":"Qi","year":"2019","journal-title":"Politeh. Univ. Buchar. Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.cam.2017.07.013","article-title":"Some properties of central Delannoy numbers","volume":"328","author":"Qi","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.jnt.2015.07.021","article-title":"Two closed forms for the Bernoulli polynomials","volume":"159","author":"Qi","year":"2016","journal-title":"J. Number Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1515\/tmj-2017-0053","article-title":"A closed form for the Stirling polynomials in terms of the Stirling numbers","volume":"10","author":"Qi","year":"2017","journal-title":"Tbilisi Math. J."},{"key":"ref_18","first-page":"297","article-title":"A determinantal expression and a recurrence relation for the Euler polynomials","volume":"16","author":"Qi","year":"2017","journal-title":"Adv. Appl. Math. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"153","DOI":"10.2298\/AADM170405004Q","article-title":"A diagonal recurrence relation for the Stirling numbers of the first kind","volume":"12","author":"Qi","year":"2018","journal-title":"Appl. Anal. Discrete Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"121","DOI":"10.5937\/KgJMath1701121F","article-title":"Explicit and recursive formulas, integral representations, and properties of the large Schr\u00f6der numbers","volume":"41","author":"Qi","year":"2017","journal-title":"Kragujevac J. Math."},{"key":"ref_21","first-page":"167","article-title":"Expressing the generalized Fibonacci polynomials in terms of a tridiagonal determinant","volume":"72","author":"Qi","year":"2017","journal-title":"Matematiche (Catania)"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Qi, F., and Guo, B.-N. (2016). Some determinantal expressions and recurrence relations of the Bernoulli polynomials. Mathematics, 4.","DOI":"10.3390\/math4040065"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"81","DOI":"10.22342\/jims.23.1.274.81-87","article-title":"Two nice determinantal expressions and a recurrence relation for the Apostol\u2013Bernoulli polynomials","volume":"23","author":"Qi","year":"2017","journal-title":"J. Indones. Math. Soc."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Qi, F., Lim, D., and Guo, B.-N. (2019). Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations. Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat. RACSAM, 113.","DOI":"10.1007\/s13398-017-0427-2"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1126","DOI":"10.1186\/s40064-016-2793-1","article-title":"Some properties of the Catalan\u2013Qi function related to the Catalan numbers","volume":"5","author":"Qi","year":"2016","journal-title":"SpringerPlus"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s13398-018-0494-z","article-title":"Some identities for a sequence of unnamed polynomials connected with the Bell polynomials","volume":"113","author":"Qi","year":"2019","journal-title":"Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Math. RACSAM"},{"key":"ref_27","first-page":"15","article-title":"Two explicit formulas of the Schr\u00f6der numbers","volume":"16","author":"Qi","year":"2016","journal-title":"Integers"},{"key":"ref_28","first-page":"282","article-title":"Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers","volume":"8","author":"Qi","year":"2016","journal-title":"Acta Univ. Sapientiae Math."},{"key":"ref_29","first-page":"857","article-title":"Several formulas for special values of the Bell polynomials of the second kind and applications","volume":"7","author":"Qi","year":"2017","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_30","unstructured":"Qi, F., Wang, J.-L., and Guo, B.-N. (2019). A determinantal expression for the Fibonacci polynomials in terms of a tridiagonal determinant. Bull. Iranian Math. Soc., 45."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1232878","DOI":"10.1080\/23311835.2016.1232878","article-title":"A recovery of two determinantal representations for derangement numbers","volume":"3","author":"Qi","year":"2016","journal-title":"Cogent Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"7","DOI":"10.5937\/KgJMath1801007F","article-title":"A representation for derangement numbers in terms of a tridiagonal determinant","volume":"42","author":"Qi","year":"2018","journal-title":"Kragujevac J. Math."},{"key":"ref_33","first-page":"1909","article-title":"Some properties of the Bernoulli numbers of the second kind and their generating function","volume":"55","author":"Qi","year":"2018","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1007\/s13398-017-0401-z","article-title":"Closed forms for derangement numbers in terms of the Hessenberg determinants","volume":"112","author":"Qi","year":"2018","journal-title":"Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat. RACSAM"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1186\/s13660-015-0738-9","article-title":"Several closed expressions for the Euler numbers","volume":"2015","author":"Wei","year":"2015","journal-title":"J. Inequal. Appl."},{"key":"ref_36","unstructured":"Bourbaki, N. (2004). Functions of a Real Variable, Elementary Theory, Springer. Translated from the 1976 French Original by Philip Spain, Elements of Mathematics (Berlin)."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/782\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:57:55Z","timestamp":1760187475000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/782"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,12]]},"references-count":36,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["sym11060782"],"URL":"https:\/\/doi.org\/10.3390\/sym11060782","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,6,12]]}}}