SOME PROPERTIES AND AN APPLICATION OF MULTIVARIATE EXPONENTIAL POLYNOMIALS

Feng Qi (1, 2, 3) , Da-Wei Niu (4) , Dongkyu Lim (5) , Bai-Ni Guo (3)

1 Tianjin Polytechnic University
2 Inner Mongolia University for Nationalities
3 Henan Polytechnic University
4 Henan University of Animal Husbandry and Economy
5 Andong National University

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 Fàa 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 multi-variate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

Mots clés: Faa di Bruno formula; logarithmic convexity; explicit formula; logarithmic concavity; completely monotonic function; inversion formula; recurrence relation; determinantal inequality; exponential polynomial; Touchard polynomial; Bell number; multi-order exponential number; property; multivariate exponential polynomial; inversion theorem; Bell polynomial of the second kind; Stirling number; white noise distribution theory; product inequality

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]; Analyse classique [math.CA]; Probabilités [math.PR]; Statistiques [math.ST]

Feng Qi, Da-Wei Niu, Dongkyu Lim, Bai-Ni Guo. SOME PROPERTIES AND AN APPLICATION OF MULTIVARIATE EXPONENTIAL POLYNOMIALS. Mathematical Methods in the Applied Sciences, 2020, 43 (6), pp.2967--2983. ⟨10.1002/mma.6095⟩. ⟨hal-01745173v2⟩