A $q$-analogue of the Biperiodic Fibonacci Sequence

The Fibonacci sequence has been generalized in many ways. One of them is defined by the relation $t_n=at_{n-1}+t_{n-2}$ if $n$ is even, $t_n=bt_{n-1}+t_{n-2}$ if $n$ is odd, with initial values $t_0=0$ and $t_1=1$, where $a$ and $b$ are positive integers. This sequence is called biperiodic Fibonacci sequence. In this paper, we introduce a $q$-analogue of this sequence. We prove several identities of $q$-analogues of the Fibonacci sequence. We give algebraic and combinatorial proofs.