Abstract
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.
Highlights
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 Levy-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
We can directly verify by definition that ln(1 + t) is a Bernstein function
Comparing the integral representation (1.4), which is the Levy-Khintchine representation of ln(1 + t), with (1.2) shows that ln(1 + t) is a Bernstein function
Summary
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 Levy-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. 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 Levy-Khintchine representation Integral representations for the logarithmic function and its reciprocal.
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