Abstract

In 1858 Cayley considered a particular kind of tridiagonal determinants (or continuants). By a direct inspection of the first cases, he conjectured an identity expressing these determinants in terms of certain other determinants considered by Sylvester in 1854. Then Cayley proved the conjectured identity by induction but, as he wrote, he felt unsatisfied with his proof. The main aim of this paper is to give a straightforward proof of Cayley's identity using the method of formal series. Moreover we use this method and umbral calculus techniques to obtain several other identities.

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.

Keywords: Cayley continuant; Kac matrix; Clement matrix; Mittag–Leffler polynomial; Pidduck polynomial; Meixner polynomial; Delannoy number; Catalan-like path; Lattice path; Hankel determinant; Connection constants; Linearization coefficients