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

Abstract

Abstract. 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. Multi-order logarithmic polynomialsLet g(t) = e t − 1 for t ∈ R and denote x m = (x 1 , x 2 , . . . , 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 = 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.

Integral representations for multivariate logarithmic polynomials

“…Multivariate logarithmic polynomials. In [10], the notion "multivariate logarithmic polynomials" was introduced.…”

“…In the paper [10], the author established an explicit formula, an identity, and two recurrence relations for multivariate logarithmic polynomials L m,n (x m ) 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 constructed some determinantal inequalities, product inequalities, logarithmic convexity for multivariate logarithmic polynomials L m,n (x m ) by virtue of some properties of completely monotonic functions.…”