ON MULTI-ORDER LOGARITHMIC POLYNOMIALS AND THEIR EXPLICIT FORMULAS , RECURRENCE RELATIONS

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: qifeng618@gmail.com, qifeng618@hotmail.com. 2010 Mathematics Subject Classification. Primary 11B83; Secondary 11A25, 11B73, 11C08, 11C20, 15A15, 26A24, 26A48, 26C05, 26D05, 33B10, 34A05.

According to the monograph [6, pp. 140-141], logarithmic polynomials L n was defined by ln where g 0 = G(a) = 1, g n = G (n) (a) for n ∈ N, and G(x) is an infinitely differentiable function at x = a.In other words, the logarithmic polynomials L n are expressions for the nth derivative of ln G(x) at the point x = a.See also [23,Section 5.2].
We now introduce two new notions "multi-order logarithmic numbers" and "multiorder logarithmic polynomials".Definition 1.1.For x k ∈ R and 1 ≤ k ≤ m, denote x m = (x 1 , x 2 , . . ., x m−1 , x m ).Let h(t) = ln(1 + t) for t > −1.Define H(t; x m ) and L m,n (x m ) by We call L m,n (x m ) higher order logarithmic polynomials, logarithmic polynomials of order m, m-variate logarithmic polynomials, multivariate logarithmic polynomials, logarithmic polynomials of m variables x 1 , x 2 , . . ., x m , multi-order logarithmic polynomials alternatively.When . . ., 1) by L m,n and call them higher order logarithmic numbers, logarithmic numbers of order m, and multi-order logarithmic numbers alternatively.
It is not difficult to see that each of the generating functions G(t; x m ) and H(t; x m ) is an inverse function of another one.
From the limit lim t→0 H(t; x m ) = 0, it follows that L m,0 (x m ) = 0 for m ≥ 1.By the software Mathematica, we can obtain which imply that the first few 1-order logarithmic numbers L Basing on the above concrete examples, we guess that all multi-order logarithmic numbers (−1) n−1 L m,n are positive integers and that all multi-order logarithmic polynomials (−1) n−1 L m,n (x m ) are positive integer polynomials of variables x 1 , x 2 , . . ., x m with degree m × n for m, n ∈ N.This guess will be verified in next section.

Recurrence relations and explicit formulas
In this section, we present two recurrence relations, one explicit formula, and one identity for multi-order logarithmic polynomials L m,n (x m ).
Theorem 2.1.For m, n ∈ N, multi-order logarithmic polynomials L m,n (x m ) can be computed by the forward recurrence relation by the backward recurrence relation by the explicit formula and by the identity for 0 = n, where s(n, k) and S(n, k) for n ≥ k ≥ 0 respectively stand for the Stirling numbers of the first and second kinds which can be generated respectively by Proof.In combinatorics, the Bell polynomials of the second kind B n,k (x n−k+1 ) are defined by and satisfy two identities ) see [6,p. 135,Theorem B], where a and b are any complex numbers.The Faà di Bruno formula for higher order derivatives of composite functions can be described in terms of the Bell polynomials of the second kind B n,k (x n−k+1 ) by (2.8) By Definition 1.1 and the formulas (2.7), (2.5), and (2.6) in sequence, we have For m ≥ 2, the generating function H(t; x m ) can be rewritten as Hence, making use of the formulas (2.7), (2.5), and (2.6) in sequence gives Computing the recurrence relation (2.1) results in The explicit formula (2.3) thus follows.
Applying the above mentioned inversion theorem inductively and recursively to the backward recurrence relation (2.2) produces which can be rearranged as (2.4).This can also be derived from applying the inversion theorem in (2.8) to the explicit formula (2.3).The proof of Theorem 2.1 is complete.
Remark 2.2.When taking 1, we can derive two recurrence relations, one explicit formula, and one identity for multi-order logarithmic numbers L m,n .

Inequalities
In this section, we will construct some determinantal and product inequalities for multi-order logarithmic polynomials L m,n (x m ) and derive the logarithmic convexity and logarithmic concavity for the sequences {L m,n (x)} n≥1 and Lm,n(x) n! n≥1 respectively.
Theorem 3.1.Let q ≥ 1 be a positive integer, let |e ij | q denote a determinant of order q with elements e ij , and let (1) If a i for 1 ≤ i ≤ q are non-negative integers, then (2) If a = (a 1 , a 2 , . . ., a q ) and b = (b 1 , b 2 , . . ., b q ) are non-increasing q-tuples of non-negative integers such that Proof.Recall from [15, Chapter XIII], [25, Chapter 1], and [28, Chapter IV] that an infinitely differentiable function f is said to be completely monotonic on an interval I if it satisfies (−1) k f (k) (x) ≥ 0 on I for all k ≥ 0. Recall also from [25, p. 21 In [14] and [15, p. 367], it was proved that if f (t) is completely monotonic on [0, ∞), then f (ai+aj ) (t) q ≥ 0 and (−1) ai+aj f (ai+aj ) (t) q ≥ 0. ( Applying f (t) to the derivative H t (t; x m ) in (3.5), taking the limit t → 0 + , and considering the equation The product inequality (3.2) thus follows.Theorem 3.1 is proved.