Number of roots of the continuant over a finite local ring

The aim of this article is to obtain a formula giving, for a positive integer $n$, the number of roots of the $n^{th}$ continuant polynomial over a finite local ring. In particular, we will give counting formulae for the roots of the continuant over the local rings ${\mathbb{F}}_q$, $\mathbb{Z}\mathrm{/}{p}^m\mathbb{Z}$ and $\frac{{\mathbb{F}}_q[X]}{<X^m>}$. To conclude, the methods used for the continuant will allow us to give a new and short proof of the counting formulae for $\lambda$-quiddities (which are the solutions of a matrix equation appearing in the study of Coxeter’s friezes) over the rings $\mathbb{Z}\mathrm{/}{p}^m\mathbb{Z}$.