双周期Fibonacci序列的$q$-模拟
A $q$-analogue of the Biperiodic Fibonacci Sequence
摘要：
斐波那契数列已经在许多方面得到了推广。其中之一由关系式$t_n=at_{n-1}+t_{n-2}$（如果$n$是偶数）和$t_n=bt_{n-1}+t_{n-2}$（如果$n$是奇数）定义，初始值为$t_0=0$和$t_1=1$，其中$a$和$b$是正整数。这个数列称为双周期斐波那契数列。在本文中，我们介绍了这个序列的$q$-类似物。我们证明了几个身份的$q$-类似物的斐波那契序列。我们给出了代数和组合的证明。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
