Umbral operators for Cayley and Sylvester continuants

Abstract

We study the main umbral operators J, M and N associated with the Cayley continuants U(?)n(x) and the generalized Sylvester continuants H(?)n(x) = U(?+n)n(x). In particular, we obtain their representation in terms of the differential operator Dx and the shift operator E. Then, by using these representations, we obtain some combinatorial and differential identities for the continuants U(?)n(x) and H(?)n(x).