Integral representations for multivariate logarithmic polynomials

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.

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]…”

Notes on two kinds of special values for the Bell polynomials of the second kind

“…In short, we obtain where 0 0 D 1 and p q D 0 for q > p 0. For detailed information on applications of the formula (2.11), please refer to the papers [4][5][6][7][8][9][11][12][13][14] and closely related references therein. Then it follows from (2.7), (2.9), (2.10), and (2.11) that, when denoting u D u.t / D 1 t 2 ,…”

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.…”