Abstract
In the paper, the author introduces a new notion “multivariate logarithmic polynomial”, establishes two recurrence relations, an explicit formula, and an identity for multivariate logarithmic polynomials by virtue of the Faà di Bruno formula and two identities for the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, some product inequalities, and logarithmic convexity for multivariate logarithmic polynomials by virtue of some properties of completely monotonic functions.
Highlights
We call Lm,n(xm) higher order logarithmic polynomials, logarithmic polynomials of order m, m-variate logarithmic polynomials, multivariate logarithmic polynomials, logarithmic polynomials of m variables x1, x2, . . . , xm, multi-order logarithmic polynomials alternatively
We introduce two new notions “multi-order logarithmic numbers” and “multiorder logarithmic polynomials”
Basing on the above concrete examples, we guess that all multi-order logarithmic numbers (−1)n−1Lm,n are positive integers and that all multi-order logarithmic polynomials (−1)n−1Lm,n(xm) are positive integer polynomials of variables x1, x2, . . . , xm with degree m × n for m, n ∈ N
Summary
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]. Basing on the above concrete examples, we guess that all multi-order logarithmic numbers (−1)n−1Lm,n are positive integers and that all multi-order logarithmic polynomials (−1)n−1Lm,n(xm) are positive integer polynomials of variables x1, x2, .
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