Higher order bell polynomials and the relevant integer sequences

The Bell polynomials [3] are a mathematical tool for representing the nth derivative of a composite function. They are strictly related to partitions [1], [2], [21]. Several applications of the classical Bell polynomials have been considered in [5], [7], [9] (in connection with [22]), [13], [14]. Some generalized forms of Bell polynomials appeared in literature, see e.g. [11], [20]. Further generalizations can be found in [15], [16], and for the multidimensional case in [6], [19]. In particular, in [15], the higher order Bell polynomials and their main properties were introduced and recently, in [18], a recursion formula for the polynomial coefficients An,k of the classical Bell polynomials was derived. This last result allows to compute the complete Bell polynomials Bn and the relevant Bell numbers bn, for every integer n. In this article, after recalling this theory, and by using a more compact notation borrowed from [6], we prove the recurrence relation formula for the polynomial