Some properties and an application of multivariate exponential polynomials

Abstract

In the paper, the authors introduce a notion “multivariate exponential polynomials” which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multivariate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

Special values of the Bell polynomials of the second kind for some sequences and functions

“…For more information on applications of the identity (1.3), please read the papers [11,12,13,15,18,30,32,35,45,50,51,60,63,73,74,79,84,100,101,106,107,109] and closely related references therein.…”

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]…”

Three closed forms for convolved Fibonacci numbers

“…By the way, in the papers [8,20,34,39,43,48,54,50,55,56,59,64] and closely-related references therein, there are some new results and applications of special values of the Bell polynomials of the second kind B n,k . Remark 3.…”