Callan-like identities

In 1998, D. Callan obtained a binomial identity involving the derangement numbers. In this paper, by using the theory of formal series, we extend such an identity to the generalized derangement numbers. Then, by using the same technique, we obtain other identities of the same kind for the generalized arrangement numbers, the generalized Laguerre polynomials, the generalized Hermite polynomials, the generalized exponential polynomials and the generalized Bell numbers, the hyperharmonic numbers, the Lagrange polynomials and the Gegenbauer polynomials.