A $q$-analogue of the Biperiodic Fibonacci Sequence

TL;DR

This paper introduces a $q$-analogue of the biperiodic Fibonacci sequence, extending the classical sequence with a new parameter and providing multiple identities with algebraic and combinatorial proofs.

Contribution

The paper presents the first $q$-analogue of the biperiodic Fibonacci sequence, including new identities and proof techniques.

Findings

Established several identities for the $q$-biperiodic Fibonacci sequence

Provided algebraic proofs of the identities

Presented combinatorial proofs complementing the algebraic approach

Abstract

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.