Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 840345, 11 pages
http://dx.doi.org/10.1155/2012/840345
Review Article

Incomplete Bivariate Fibonacci and Lucas 𝑝 -Polynomials

Dursun Tasci,1 Mirac Cetin Firengiz,2 and Naim Tuglu1

1Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, 06500 Ankara, Turkey
2Department of Mathematics, Faculty of Education, Başkent University, Baglica, 06810 Ankara, Turkey

Received 26 November 2011; Accepted 8 February 2012

Academic Editor: Gerald Teschl

Copyright © 2012 Dursun Tasci 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

We define the incomplete bivariate Fibonacci and Lucas 𝑝 - polynomials. In the case 𝑥 = 1 , 𝑦 = 1 , we obtain the incomplete Fibonacci and Lucas 𝑝 - numbers. If 𝑥 = 2 , 𝑦 = 1 , we have the incomplete Pell and Pell-Lucas 𝑝 - numbers. On choosing 𝑥 = 1 , 𝑦 = 2 , we get the incomplete generalized Jacobsthal number and besides for 𝑝 = 1 the incomplete generalized Jacobsthal-Lucas numbers. In the case 𝑥 = 1 , 𝑦 = 1 , 𝑝 = 1 , we have the incomplete Fibonacci and Lucas numbers. If 𝑥 = 1 , 𝑦 = 1 , 𝑝 = 1 , 𝑘 = ⌊ ( 𝑛 − 1 ) / ( 𝑝 + 1 ) ⌋ , we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas 𝑝 - polynomials are given.