A New Generalization of Pell-Lucas Numbers (Bi-Periodic Pell-Lucas Sequence) - Uygun, Sukran; Karatas, Hasan (Communications in Mathematics and Applications, 2019)
In this study, we bring into light, a new generalization of the Jacobsthal Lucas numbers, which shall also be called the bi-periodic Jacobsthal Lucas sequence as\[Q_{n}=\begin{cases}2bQ_{n-1}+Q_{n-2},&\text{if} \ n \ \text{is even} \\2aQ_{n-1}+Q_{n-2},&\text{if} \ n \ \text{is odd}\end{cases}\quad n\geq 2,\]with initial conditions \(Q_{0}=2\), \(Q_{1}=a\). The Binet formula as well as the generating function for this sequence are given. The convergence properties of the consecutive terms of this sequence are also examined after which the well known Cassini, Catalans and the D'Ocagne's identities as well as some related summation formulas are also given.
Keywords: Bi-periodic Pell sequence; Pell-Lucas sequence; Generating function; Binet formula