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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.

On higher order generalized geometric polynomials with shifted parameters