Abstract
The derangement number [Formula: see text] is the number of fixed point free permutations on a set of [Formula: see text] elements and the derangement polynomial [Formula: see text]) is a natural extension of the derangement number [Formula: see text]. The aim of this paper is to introduce [Formula: see text]-derangement numbers and polynomials, which are [Formula: see text]-analogs of the derangement numbers and polynomials, and to investigate their connection with some other [Formula: see text]-special numbers and polynomials. In more detail, we derive explicit expressions and recurrence relations for the [Formula: see text]-derangement numbers and polynomials. Further, we obtain some identities involving such polynomials and numbers and other special [Formula: see text]-polynomials and numbers, which include [Formula: see text]-Bell polynomials, [Formula: see text]-analogs of Fubini polynomials and [Formula: see text]-Stirling numbers of the second kind.
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