We prove two general results on generalized symmetries for equations of the form ut = um + f(u, u1, . . . , um−1), where f is a formal (differential) power series starting with terms that are at least quadratic. The first result states that any higher order symmetry must be also a differential polynomial if f is a differential polynomial of order less than m − 1. The method is to estimate the o...

Abstract In the present paper we introduce some expansions, based on the falling factorials, for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Faá di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of con...

A Faà di Bruno type Hopf algebra is developed for a group of integral operators known as Fliess operators, where operator composition is the group product. The result is applied to analytic nonlinear feedback systems to produce an explicit formula for the feedback product, that is, the generating series for the Fliess operator representation of the closed-loop system written in terms of the gen...

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second...

In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion form...

In the paper, by virtue of Faà di Bruno formula, with aid some properties Bell polynomials second kind, and means a general formula for derivatives ratio between two differentiable functions, authors establish explicit, determinantal, recurrent formulas generalized Eulerian polynomials.

In this paper, the author presents several closed forms and determinantal expressions involving Stirling numbers of second kind for higher-order Bernoulli Euler polynomials by applying Faà di Bruno formula some properties Bell polynomials.

Abstract In this paper, with the aid of Faà di Bruno formula and by virtue properties Bell polynomials second kind, authors define a kind notion degenerate Narumi numbers polynomials, establish explicit formulas for derive (degenerate) Cauchy numbers.