Journal of Mathematics Research; Vol. 10, No. 3; June 2018 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education 35 Applications of Pell Polynomials in Rings Yasemin TAŞYURDU 1 , Devran ÇİFÇİ 2 , Ömür DEVECİ 3 1,2 Erzincan University, Department of Mathematics, Faculty of Science and Art, 24000 Erzincan, Turkey 3 Kafkas University, Department of Mathematics, Faculty of Science and Art, 36100 Kars, Turkey Correspondence: Yasemin TAŞYURDU, Erzincan University, Department of Mathematics, Faculty of Science and Art, 24000 Erzincan, Turkey. Received: November 7, 2017 Accepted: : November 27, 2018 Online Published: March 26, 2018 doi:10.5539/jmr.v10n3p35 URL: https://doi.org/10.5539/jmr.v10n3p35 Abstract In this paper, we study the Pell polynomials according to modulo m where 2 = 2 + 1 and various properties of these sequences are obtained. Also, Pell polynomials to the ring of complex numbers was defined. We define the Pell Polynomial-type orbits (,) () = * + where be a 2-generator ring and (, ) is a generating pair of the ring . Furthermore, we obtain the periods of the Pell Polynomial-type orbits (,) () in finite 2-generator rings of order 2 . 2000 Mathematics Subjet Classification: 11B37, 11B83, 16P10 Keywords: Pell polynomials, Period, Ring 1. Introduction Integers sequences, such as Fibonacci, Lucas, Pell and Jacobsthal have been an intriguing topic for many years in Applied Mathematics. Many authors are dedicated to study this sequence, such as the work in (Deveci, 2015; Deveci & Saraçoğlu Eskiyapar, 2016) and many other (Knox, 1992; Kılıç & Taşçı 2005). Fibonacci and Pell numbers are the most known numbers. Fibonacci sequence is defined by the equation = −1 + −2 where = −1 + −2 with the initial values 0 =0, 1 =1, ≥2 and are the terms of the sequence 0,1,1,2,3,5,8,13 … The Pell sequence is defined recursively by the equation +2 = 2 −1 + for ≥0 where 0 =0 and 1 =1 . The Pell sequence is 0,1,2,5,12,29,70,169 …. Most of the study of Fibonacci polynomials and Pell polynomials are applications in groups. D. D. Wall proposed the nation of Wall number of the Fibonacci sequence in 1960 and obtained many properties and theorems about these numbers (Wall 1960). Wilcox extend the problem to abelian groups (Wilcox 1986). Fibonacci polynomials is defined as, () = −1 () + −2 () ; ≥2 with 0 () = 0, 1 () = 1 (11) Lucas polynomials is defined as, () = −1 () + −2 (); ≥2 with 0 () = 2, 1 () = 1 (12) Pell polynomials is defined as, () = 2 −1 () + −2 (); ≥2 with 0 () = 0, 1 () = 1 (13) Note the Pell polynomials are generated by matrix =. 2 1 1 0 /, () =( +1 () () () −1 () ) (1.4) which can be proved by mathematical induction (Kılıç & Taşçı 2005). Pell-Lucas polynomials is defined as, () = 2 −1 () + −2 (); ≥2 with 0 () = 2, 1 () = 2 (15) D. J. DeCarli described the generalized Fibonacci sequences on an arbitrary ring in 1970 (DeCarli 1970). R. G. Buschman, A. F. Horadam and N. N. Vorobyov considered the set of integers for the special cases of these rings (Buschman, 1963;