On Multi-order Logarithmic Polynomials and Their Explicit Formulas, Recurrence Relations, and Inequalities

In the paper, the author introduces the notions “multi-order logarithmic numbers” and “multi-order logarithmic polynomials”, establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faà di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions. 1. Multi-order logarithmic polynomials Let g(t) = e − 1 for t ∈ R and denote xm = (x1, x2, . . . , xm−1, xm) for xk ∈ R and 1 ≤ k ≤ m. Recently, the quantities Qm,n(xm) were defined by G(t;xm) = exp(x1g(x2g(· · ·xm−1g(xmg(t)) · · · ))) = ∞ ∑ n=0 Qm,n(xm) t n! and were called the Bell-Touchard polynomials [22]. When m = 1 and x1 = 1, the quantities Q1,n(1) = Bn were called the Bell numbers [1, 5, 8, 17, 18] or exponential numbers [2] and were generalized and applied [1]. When m = 1 and x1 = x is a variable, the quantities Q1,n(x) = Bn(x) = Tn(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 Qm,n(x) were investigated. For more information on this topic, please refer to [22] and closely related references therein. E-mail address: [email protected], [email protected]. 2010 Mathematics Subject Classification. Primary 11B83; Secondary 11A25, 11B73, 11C08, 11C20, 15A15, 26A24, 26A48, 26C05, 26D05, 33B10, 34A05.