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

Authors : Feng Qi, BaiNi Guo

Pages : 191-194

Doi:10.36753/mathenot.566448

View : 38 | Download : 9

Publication Date : 2019-10-15

Article Type : Research Paper

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 Tkinsert ignore into journalissuearticles values(x); for k ≥ 0 can be generated [8– 10, 15, 17] by e xinsert ignore into journalissuearticles values(e t−1); = X∞ k=0 Tkinsert ignore into journalissuearticles values(x); t k k! = 1 + xt + 1 2 xinsert ignore into journalissuearticles values(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 + · · · insert ignore into journalissuearticles values(1.1); The polynomials Tkinsert ignore into journalissuearticles values(x); for k ≥ 0 are also called [11, 18] the Touchard polynomials or the exponential polynomials. It is clear that Tkinsert ignore into journalissuearticles values(1); = Bk. The central factorial numbers of the second kind Tinsert ignore into journalissuearticles values(n, k); for n ≥ k ≥ 0 can be generated [1, 6] by 1 k! 2 sinh t 2 k = X∞ n=k Tinsert ignore into journalissuearticles values(n, k); t n n! , where sinh t = e t − e −t 2 insert ignore into journalissuearticles values(1.2); is the hyperbolic sine function. The central Bell polynomials B insert ignore into journalissuearticles values(c); k insert ignore into journalissuearticles values(x); for k ≥ 0 can be generated [5] by exp 2x sinh t 2 = X∞ k=0 B insert ignore into journalissuearticles values(c); k insert ignore into journalissuearticles values(x); t k k! . 
Keywords : Bell polynomial, central factorial number of the second kind, central Bell polynomial, Bell polynomial of the second kind, Faà di Bruno formula

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