Cayley continuants

Cayley continuants appear in several contexts and in particular in enumerative combinatorics. Mittag-Leffler polynomials, Meixner polynomials of the first kind, the falling and the raising factorials are just few instances of these continuants. They can be interpreted in terms of weighted permutations. Moreover, as we prove in this paper, they also appear in the context of Hankel determinants generated by certain Catalan-like numbers.