INTEGRAL REPRESENTATIONS FOR MULTIVARIATE LOGARITHMIC POLYNOMIALS

In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a Lévy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.

Chapter XIII], [13,Chapter 1], and [14, 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. Theorem 12b in [14] reads that a necessary and sufficient condition that f (t) should be completely monotonic for 0 < t < ∞ is that where α(s) is non-decreasing and the integral converges for 0 < s < ∞.
Recall also from [13, p. 21, Definition 3.1] that a nonnegative function f : (0, ∞) → R is a Bernstein function if its first derivative f is completely monotonic on (0, ∞). Theorem 3.2 in [13] states that, a function f : (0, ∞) → [0, ∞) is a Bernstein function if and only if it admits the Lévy-Khintchine representation and µ is called the Lévy measure on (0, ∞) satisfying ∞ 0 min{1, t} d µ(t) < ∞. 1.2. Integral representations for the logarithmic function and its reciprocal. We can directly verify by definition that ln(1 + t) is a Bernstein function.
Then it is clear that 1 ln(1+t) is a completely monotonic function on (0, ∞). In [12, p. 996], it was presented that where t > 0 and Γ(z) is the classical gamma function which can be defined by Then it is easy to see that and has the representation t . Theorem 3.1 in [5,6,7] reads that, if φ is a Bernstein function, then (1.7) By virtue of (1.6) and (1.7), we see that the complex function ln(1 + z) for z ∈ C \ (−∞, −1] has the integral representation The formula (1.8) can also be derived from taking a = b = g = 1 and c = z in 1.3. Multivariate logarithmic polynomials. In [10], the notion "multivariate logarithmic polynomials" was introduced.
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 x 1 = x 2 = · · · = x m−1 = x m = 1, we denote L m,n (1, . . . , 1) by L m,n and call them higher logarithmic numbers, logarithmic numbers of order m, and multi-order logarithmic numbers alternatively.
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.
Naturally we pose a question: does the generating function H(t; x m ) have similar properties to the above ones for ln(1+t) and its reciprocal? can these properties for H(t; x m ) be applied to derive corresponding properties of multivariate logarithmic polynomials L m,n (x m )?

Integral representations for multivariate logarithmic polynomials
In this section, when x 1 , x 2 , . . . , x m > 0 and m ∈ N, by induction and recursively, we prove that the generating function H(t; x m ) and its reciprocal for s m = (s 0 , s 1 , . . . , s m ). Consequently, the multivariate logarithmic polynomials L m,n (x m ) for m, n ∈ N can be represented by the integral Furthermore, it follows that and, inductively, that which can be rearranged as (2.1) and again confirm that the generating function H(t; x m ) is a Bernstein function. Considering (1.9) in Definition 1.1, differentiating on both sides of (2.1) with respect to t, and taking the limit t → 0 + derive (2.3) readily. The proof of Theorem 2.1 is complete. Proof. Theorem 3.7 in [13, p. 27] states that f is a Bernstein function if and only if g •f is completely monotonic for every completely monotonic function g. Therefore, by induction, for x 1 , . . . , x m > 0, the function 1 H(t;xm) is completely monotonic with respect to t ∈ (0, ∞).
On the other hand, making use of (1.5), we obtain Furthermore, it follows that and, by induction, that for t, x > 0. Taking m = 1 in (2.4) or retrospecting its proof reduces to We observe that these three integrals between (2.5) and (2.7) are respectively single, double, and triple and that the integrand in (2.5) is elementary but the integrands in (2.6) and (2.7) are not elementary. Consequently, we naturally pose a question: can one find out general integral representations, which are single integrals and whose integrands are elementary, for the generating function H(t; x m ) and multivariate logarithmic polynomials L m,n (x m )? One way to answer this question is to explicitly and elementarily express [H(si; x m )] and to apply the formula (1.6). where U m (x 1 , . . . , x m ; s) and V m (x 1 , . . . , x m ; s) satisfy the recurrence relations