International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 709386, 21 pages
Research Article

On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

1Department of Mathematics and Computer Science, Illinois Wesleyan University, Bloomington, IL 61702, USA
2Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154, USA

Received 7 July 2009; Accepted 14 September 2009

Academic Editor: Misha Rudnev

Copyright © 2009 Tian-Xiao He and Peter J.-S. Shiue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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-Humbert 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.