Relations among Bell polynomials, central factorial numbers, and central Bell polynomials

Abstract

In the note, by virtue of the Fa\`a di Bruno formula and two identities for the Bell polynomials of the second kind, the authors derive three relations among the Bell polynomials, central factorial numbers of the second kind, and central Bell polynomials. The Bell numbers Bk for k ≥ 0 can be generated [4, 7, 12] byee t−1 =X∞ k=0Bktk k!= 1 + t + t 2 +5 6 t 3 + 5 8 t 4 + 13 30 t 5 + 203 720 t 6 + 877 5040 t 7 + · · · As a generalization of the Bell numbers Bk for k ≥ 0, the Bell polynomials Tk(x) for k ≥ 0 can be generated [8– 10, 15, 17] by e x(e t−1) = X∞ k=0 Tk(x) t k k! = 1 + xt + 1 2 x(x + 1)t 2 + 1 6 x x 2 + 3x + 1 t 3 + 1 24 x x 3 + 6x 2 + 7x + 1 t 4 + 1 120 x x 4 + 10x 3 + 25x 2 + 15x + 1 t 5 + · · · (1.1) The polynomials Tk(x) for k ≥ 0 are also called [11, 18] the Touchard polynomials or the exponential polynomials. It is clear that Tk(1) = Bk. The central factorial numbers of the second kind T(n, k) for n ≥ k ≥ 0 can be generated [1, 6] by 1 k! 2 sinh t 2 k = X∞ n=k T(n, k) t n n! , where sinh t = e t − e −t 2 (1.2) is the hyperbolic sine function. The central Bell polynomials B (c) k (x) for k ≥ 0 can be generated [5] by exp 2x sinh t 2 = X∞ k=0 B (c) k (x) t k k! .