Abstract
In this paper we investigate the generalized Pell numbers of order r ≥ 2 through the properties of their related fundamental system of generalized Pell numbers. That is, the generalized Pell number of order r ≥ 2; are expressed as a linear combination of a fundamental system of generalized Pell numbers. The properties of this fundamental system are examined and results can be established for generalized Pell numbers of order r ≥ 2. Some identities and combinatorial results are established. Moreover, the analytic study of the fundamental system of generalized Pell is provided. Furthermore, the generalized Pell Cassini identity type is provided.
Highlights
The usual sequence of Pell numbers (Pn)n≥0 is defined by the initial conditions P0 = 0, P1 = 1, and the recurrence relation Pn+1 = 2Pn + Pn−1, for n ≥ 1
We explore the family of generalized Pell numbers (1.1), through the properties of the set Pr
Let (Pn)n≥0 be a sequence of generalized Pell numbers defined by the recursive relation (1.1), and whose initial conditions are P0 = α0, P1 = α1, . . . , Pr−1 = αr−1, andn≥0 be the sequence defined by wn = α0Pn(1) + α1Pn(2) + · · · + αr−1Pn(r), for every n ≥ 0
Summary
Let consider the set Pr = {(Pn(s))n≥0, 1 ≤ s ≤ r} of sequences of generalized Pell numbers (Pn(s))n≥0 defined as follows, r−1. Let consider the case r = 3, Theorem 2.1 implies that for the set P3 = {(Pn(s))n≥0, 1 ≤ s ≤ 3} of the basic sequences of generalized Pell numbers (Pn(s))n≥0, we have, Pn(+1)1 = Pn(3) for n ≥ 0, or equivalently Pn(1) = Pn(−3) for n ≥ 1, Pn(2) = Pn(1) + Pn(−1)1 = Pn(−3)1 + Pn(−3), for every n ≥ 2. Let (Pn)n≥0 be a sequence of generalized Pell numbers defined by the recursive relation (1.1), and whose initial conditions are P0 = α0, P1 = α1, .
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
