Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2011, Article ID 623456, 21 pages doi:10.1155/2011/623456 Research Article On Generalized Bell Polynomials Roberto B. Corcino 1 and Cristina B. Corcino 2 1 Department of Mathematics, Mindanao State University, Marawi City 9700, Philippines 2 Department of Mathematics, De La Salle University, Manila 1004, Philippines Correspondence should be addressed to Roberto B. Corcino, rcorcino@yahoo.com Received 18 June 2011; Accepted 25 July 2011 Academic Editor: Garyfalos Papaschinopoulos Copyright q 2011 R. B. Corcino and C. B. Corcino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. It is shown that the sequence of the generalized Bell polynomials S n x is convex under some restrictions of the parameters involved. A kind of recurrence relation for S n x is established, and some numbers related to the generalized Bell numbers and their properties are investigated. 1. Introduction Hsu and Shiue 1 defined a kind of generalized Stirling number pair with three free parameters which is introduced via a pair of linear transformations between generalized factorials, viz, t | α n n k0 S ( n, k; α, β, γ )( t - γ | β ) k , ( t | β ) n n k0 S ( n, k; β, α, -γ )( t γ | α ) k , 1.1 where n ∈ N set of nonnegative integers, α, β, and γ may be real or complex numbers with α, β, γ / 0, 0, 0, and t | α n denotes the generalized factorial of the form t | α n n-1 j 0 ( t - jα ) , n ≥ 1, t | α 0 1. 1.2 In particular, t | 1 n t n with t 0 1. Various well-known generalizations were obtained by special choices of the parameters α, β, and γ cf. 1, and the generalization of some properties of the classical Stirling numbers such as the recurrence relations S ( n 1,k; α, β, γ ) S ( n, k - 1; α, β, γ ) ( kβ - nα γ ) S ( n, k; α, β, γ ) , 1.3