Abstract
In this paper, we introduce a q-analog of the bi-periodic Lucassequence, called as the q-bi-periodic Lucas sequence, and give some identities related to the q-bi-periodic Fibonacci and Lucas sequences. Also, we givea matrix representation for the q-bi-periodic Fibonacci sequence which allowus to obtain several properties of this sequence in a simple way. Moreover,by using the explicit formulas for the q-bi-periodic Fibonacci and Lucas sequences, we introduce q-analogs of the bi-periodic incomplete Fibonacci andLucas sequences and give a relation between them
Highlights
It is well-known that the classical Fibonacci numbers Fn are de...ned by the recurrence relationFn = Fn 1 + Fn 2; n 2 (1.1)with the initial conditions F0 = 0 and F1 = 1
We introduce a q-analog of the bi-periodic Lucas sequence, called as the q-bi-periodic Lucas sequence, and give some identities related to the q-bi-periodic Fibonacci and Lucas sequences
By using the explicit formulas for the q-bi-periodic Fibonacci and Lucas sequences, we introduce q-analogs of the bi-periodic incomplete Fibonacci and Lucas sequences and give a relation between them
Summary
In analogy with (1.3), Tan and Ekin [14] gave the explicit formula of the bi-periodic Lucas numbers as: pn AFn(a;1b) (q; s) + qn 2sFn(a;2b) (q; s) ; bFn(a;1b) (q; s) + qn 2sFn(a;2b) (q; s) ; if n is even ; n if n is odd with initial conditions F0(a;b) (q; s) = 0 and F1(a;b) (q; s) = 1: They derived the following equality to evaluate the q-bi-periodic Fibonacci sequence: Fn(a;b) (q; s) = nFn(a;1b) (q; qs) qsFn(a;2b) q; q2s ; (1.9) By using (1.10), they obtained the explicit formula of the q-bi-periodic Fibonacci sequence as: b n1
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
