Congruences for $q$-Lucas Numbers

Hao Pan

Abstract


For $\alpha,\beta,\gamma,\delta\in{\mathbb Z}$ and ${\rm\nu}=(\alpha,\beta,\gamma,\delta)$, the $q$-Fibonacci numbers are given by$$F_0^{{\rm\nu}}(q)=0,\ F_1^{{\rm\nu}}(q)=1\text{ and }F_{n+1}^{{\rm\nu}}(q)=q^{\alpha n-\beta}F_{n}^{{\rm\nu}}(q)+q^{\gamma n-\delta}F_{n-1}^{{\rm\nu}}(q)\text{ for }n\geq 1.$$And define the $q$-Lucas number $L_{n}^{{\rm\nu}}(q)=F_{n+1}^{{\rm\nu}}(q)+q^{\gamma-\delta}F_{n-1}^{{\rm\nu}_*}(q)$, where ${\rm\nu}_*=(\alpha,\beta-\alpha,\gamma,\delta-\gamma)$. Suppose that $\alpha=0$ and $\gamma$ is prime to $n$, or $\alpha=\gamma$ is prime to $n$. We prove that$$L_{n}^{{\rm\nu}}(q)\equiv(-1)^{\alpha(n+1)}\pmod{\Phi_n(q)}$$for $n\geq 3$, where $\Phi_n(q)$ is the $n$-th cyclotomic polynomial. A similar congruence for $q$-Pell-Lucas numbers is also established.

Keywords


$q$-Lucas number; set partition; group action

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