On multivariate logarithmic polynomials and their properties

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.

Special values of the Bell polynomials of the second kind for some sequences and functions

“…For more information on applications of the identity (1.4), please refer to [29,38,39,41,42,44,46,73,80,91,95,96] and closely related references therein.…”

Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials

“…For more information about the Bell numbers and polynomials, please refer to [2,7,9,12,19,20,31,37,39,41]…”

Some properties and an application of multivariate exponential polynomials

“…Remark In other works, multivariate logarithmic polynomials and their generating function, the inverse of the generating function , were investigated.…”