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ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 674167, 16 pages
http://dx.doi.org/10.5402/2011/674167
Research Article

Sequences of Numbers Meet the Generalized Gegenbauer-Humbert Polynomials

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 89154-4020, USA
3Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan

Received 6 July 2011; Accepted 25 August 2011

Academic Editor: W. Liu

Copyright © 2011 Tian-Xiao He et al. 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.

Abstract

Here we present a connection between a sequence of numbers generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known formulas of the Fibonacci, the Lucas, the Pell, and the Jacobsthal numbers in terms of the generalized Gegenbauer-Humbert polynomial values are given. The applications of the relationship to the construction of identities of number and polynomial value sequences defined by linear recurrence relations are also discussed.