Generalizations of the Bell Numbers and Polynomials and Their Properties

Abstract

Abstract. In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.

On Multi-Order Logarithmic Polynomials and their Explicit Formulas, Recurrence Relations, and Inequalities

“…When m = 1 and x 1 = 1, the quantities Q 1,n (1) = B n were called the Bell numbers [1,5,8,17,18] or exponential numbers [2] and were generalized and applied [1]. When m = 1 and x 1 = x is a variable, the quantities Q 1,n (x) = B n (x) = T n (x) were called the Bell polynomials [20,21], the Touchard polynomials [19,22], or exponential polynomials [3,4,7] and were applied [9,10,11,12,19]. In the paper [22], explicit formulas, recurrence relations, determinantal inequalities, product inequalities, logarithmic convexity, logarithmic concavity, and applications of Q m,n (x) were investigated.…”

“…, x m−1 , x m ) for x k ∈ R and 1 ≤ k ≤ m. Recently, the quantities Q m,n (x m ) were defined by G(t; x m ) = exp(x 1 g(x 2 g(· · · x m−1 g(x m g(t)) · · · ))) = ∞ n=0 Q m,n (x m ) t n n! and were called the Bell-Touchard polynomials [22]. When m = 1 and x 1 = 1, the quantities Q 1,n (1) = B n were called the Bell numbers [1,5,8,17,18] or exponential numbers [2] and were generalized and applied [1].…”

“…When m = 1 and x 1 = x is a variable, the quantities Q 1,n (x) = B n (x) = T n (x) were called the Bell polynomials [20,21], the Touchard polynomials [19,22], or exponential polynomials [3,4,7] and were applied [9,10,11,12,19]. In the paper [22], explicit formulas, recurrence relations, determinantal inequalities, product inequalities, logarithmic convexity, logarithmic concavity, and applications of Q m,n (x) were investigated. For more information on this topic, please refer to [22] and closely related references therein.…”

On multivariate logarithmic polynomials and their properties

“…. , x m−1 , x m ) for x k ∈ R and 1 ≤ k ≤ m. Recently, the quantities Q m,n (x m ) were defined byand were called the Bell-Touchard polynomials [22]. When m = 1 and x 1 = 1, the quantities Q 1,n (1) = B n were called the Bell numbers [1,5,8,17,18] or exponential numbers [2] and were generalized and applied [1].…”

“…When m = 1 and x 1 = 1, the quantities Q 1,n (1) = B n were called the Bell numbers [1,5,8,17,18] or exponential numbers [2] and were generalized and applied [1]. When m = 1 and x 1 = x is a variable, the quantities Q 1,n (x) = B n (x) = T n (x) were called the Bell polynomials [20,21], the Touchard polynomials [19,22], or exponential polynomials [3,4,7] and were applied [9,10,11,12,19]. In the paper [22], explicit formulas, recurrence relations, determinantal inequalities, product inequalities, logarithmic convexity, logarithmic concavity, and applications of Q m,n (x) were investigated.…”