On the determinants and inverses of $r-$circulant matrices with biperiodic Fibonacci and Lucas numbers

Abstract

In this paper, we present a new generalization to compute determinants and inverses of $r-$circulant matrices  $\mathcal{Q}_n = circ_r \left( \left( \frac{b}{a} \right)^{\frac{\xi(0)}{2}} q_1, \left( \frac{b}{a} \right)^{\frac{\xi(1)}{2}}q_2, \ldots,  \left( \frac{b}{a} \right)^{\frac{\xi(n-1)}{2}}q_n \right) $ and $\mathcal{L}_n = circ_r \left( \left( \frac{b}{a} \right)^{\frac{\xi(1)}{2}} l_1, \left( \frac{b}{a} \right)^{\frac{\xi(2)}{2}}l_2, \ldots,  \left( \frac{b}{a} \right)^{\frac{\xi(n)}{2}}l_n \right)$ whose entries are the biperiodic Fibonacci and the biperiodic Lucas numbers, respectively. Also, we express determinants of the matrices $\mathcal{Q}_n$ and $\mathcal{L}_n$ by using only the biperiodic Fibonacci and the biperiodic Lucas numbers.