On properties of bi-periodic Fibonacci and Lucas polynomials

Bi-periodic Fibonacci matrix polynomial and its binomial transforms

“…We obtained the Cassini identity for bi-periodic Fibonacci polynomials [14]. Using the determinant of F n (a, b, x) in Theorem 2.2, again we get…”

“…Also, the polynomials have attracted the attention of some mathematicians [6,7,14]. In [14], the authors gave the bi-periodic Fibonacci polynomial as q n (a, b, x) = axq n−1 (a, b, x) + q n−2 (a, b, x) , if n is even bxq n−1 (a, b, x) + q n−2 (a, b, x) , if n is odd (1.4) which q 0 (a, b, x) = 0, q 1 (a, b, x) = 1 and a, b are nonzero real numbers and they obtained some properties of this polynomial. Hoggatt and Bicknell, in [7], defined the Fibonacci, Tribonacci, Quadranacci, r-bonacci polynomials.…”

The Hadamard Products for bi-periodic Fibonacci and bi-periodic Lucas Generating matrices

“…The special sequences and their properties have been investigated in many articles and books (see, for example [1,3,5,6,8,9], [14]- [17] and the references cited therein). The Fibonacci and Lucas numbers have attracted the attention of mathematicians because of their intrinsic theory and applications.…”

“…Many authors have generalized Fibonacci sequence in different ways. In the one of those generalizations, in [17], we define the bi-periodic Fibonacci {q n (x)} n∈N polynomial as in the form…”

“…The Binet formula for bi-periodic Fibonacci polynomial, given in [17], can also be obtained by using Q q matrix. Theorem 2.4 Let n be any integer.…”