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 nᵗʰ continuant polynomial over a finite local ring. In particular, we will give counting formulae for the roots of the continuant over the local rings Fq, Z/pᵐZ and Fq[X]<Xᵐ>. To conclude, the methods used for the continuant will allow us to give a new and short proof of the counting formulae for λ-quiddities (which are the solutions of a matrix equation appearing in the study of Coxeter's friezes) over the rings Z/pᵐZ.