Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials second kind, authors reconsider generating functions Hermite their squares, find an explicit for higher-order derivatives function polynomials, derive formulas recurrence relations squares.

Strategies of carbon (C) and nitrogen (N) utilisation are among the factors determining plant distribution. It has been argued that submersed macrophytes adapted to lower light environments are more efficient in maintaining C metabolic homeostasis due to their conservative C strategy and ability to balance C shortage. We studied how depth distributions of 12 submersed macrophytes in Lake Erhai,...

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers. E-mail addr...

We study discrete functions on equidistant and non-equidistant infinitesimal grids. We consider their difference quotients of higher order and give conditions for their near-equality to the corresponding derivatives. Important tools are nonstandard notions of regularity of higher order, and the formula of Faà di Bruno for higher order derivatives and a discrete version of it. As an application ...

where the sum runs over the partitions B1, . . . , Ba of {1, . . . , n} with cardinal |Bi| by the Faà di Bruno formula, cf. [5], [6] and references therein for background on combinatorial probability. When X is centered Gaussian, e.g. X is the Wiener integral of a deterministic function with respect to a standard Brownian motion (Bt)t∈R+ , we have κ X n = 0, n 6= 2, and (1.1) reads as Wick’s th...

Fà a di Bruno (Hopf, bi)algebras appear in several branches of mathematics and physics, and may be introduced in several ways. Here we start from exponential power series like f (t) = ∞ n=1 f n n! t n , with f 1 > 0. In view of Borel's theorem, one may regard them as local representatives of orientation-preserving diffeomorphisms of R leaving 0 fixed. On the group G of these diffeomorphisms we ...

In this paper, by introducing degenerate Fubini-type polynomials, with the help of Faà di Bruno formula and some properties partial Bell authors provide several new explicit formulas recurrence relations for polynomials numbers, associate newly defined Apostol–Bernoulli Apostol–Euler order α. These results enable one to present additional special numbers.

We establish a close relation between higher order Riccati equations and Faá di Bruno polynomial respectively Ramanujan’s differential equations connected to modular forms.

1 Department of Physics and “Nanostructured Interfaces and Surfaces” Interdepartmental Centre, Università di Torino, Torino, Italy, 2 Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, Università di Trento, Trento, Italy, 3 Center for Materials and Microsystems, Fondazione Bruno Kessler, Trento, Italy, 4 School of Engineering an...

The differential transform method is used to find numerical approximations of the solution a class certain nonlinear three-point singular boundary value problems. based on Taylor's theorem. Coefficients Taylor series are determined by constructing recurrence relation. To deal with nonlinearity problems, Faa di Bruno's formula containing partial ordinary Bell polynomials applied within transform...