{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T14:52:12Z","timestamp":1779893532566,"version":"3.53.1"},"reference-count":23,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2009,10,25]],"date-time":"2009-10-25T00:00:00Z","timestamp":1256428800000},"content-version":"vor","delay-in-days":297,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer\u2010Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.<\/jats:p>","DOI":"10.1155\/2009\/709386","type":"journal-article","created":{"date-parts":[[2009,10,25]],"date-time":"2009-10-25T15:30:05Z","timestamp":1256484605000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2"],"prefix":"10.1155","volume":"2009","author":[{"given":"Tian-Xiao","family":"He","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter J.-S.","family":"Shiue","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2009,10,25]]},"reference":[{"key":"e_1_2_6_1_2","first-page":"207","article-title":"A formula for the generating functions of powers of Horadam\u2032s sequence","volume":"30","author":"Mansour T.","year":"2004","journal-title":"The Australasian Journal of Combinatorics"},{"key":"e_1_2_6_2_2","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 A. 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