Unified approach to Carlitz and Cigler q-analogue for the bi-periodic Fibonacci and Lucas sequences

Abstract

In this study, our aim is to establish and generalize a q-analogue for the bi-periodic Fibonacci and Lucas polynomials. We introduce two types of q-analogue for the bi-periodic Lucas polynomials, namely the q-bi-periodic Lucas polynomials of the first and second kinds. We extend and unify various aspects including explicit forms, recurrence relations, generating functions, and other combinatorial properties. Moreover, we give a q-analogue of the relationship between bi-periodic Fibonacci sequence and second-order recurrences