On higher order generalized geometric polynomials with shifted parameters

We study a special class of higher order generalized geometric polynomials. Based on our combinatorial interpretation of labeled barred preferential arrangements, we prove several recursions. We also study the polynomials from a probabilistic point of view, and show how our polynomials can be written in terms of the expectation of a random descending factorial involving the negative binomial process. Using techniques of probability theory, we derive identities, in particular we extend Nelsen's theorem.