A unified approach to higher order convolutions within a certain subset of appell polynomials

We consider the subset R of Appell polynomials whose exponential generating function is given in terms of the moment generating function of a certain random variable Y. This subset contains the Hermite, Bernoulli, Apostol–Euler, and Cauchy type polynomials, as well as various kinds of their generalizations, among others. We obtain closed form expressions for higher order convolutions of Appell polynomials in the subset R. We give a unified approach mainly based on random scale transformations of Appell polynomials, as well as on a probabilistic generalization of the Stirling numbers of the second kind. Different illustrative examples, including reformulations of convolution identities already known in the literature, are discussed in detail. In such examples, the convolution identities involve the classical Stirling numbers

Combinatorial identities associated with new families of the numbers and polynomials and their approximation values

Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling's approximation for factorials, we investigate some approximation values of the special case of the numbers $Y_{n}\left( \lambda \right)$.Comment: 17 page

Woon's tree and sums over compositions

This article studies sums over all compositions of an integer. We derive a generating function for this quantity, and apply it to several special functions, including various generalized Bernoulli numbers. We connect composition sums with a recursive tree introduced by S.G. Woon and extended by P. Fuchs under the name "general PI tree", in which an output sequence $\{x_n\}$ is associated to the input sequence $\{g_n\}$ by summing over each row of the tree built from $\{g_n\}$. Our link with the notion of compositions allows to introduce a modification of Fuchs' tree that takes into account nonlinear transforms of the generating function of the input sequence. We also introduce the notion of \textit{generalized sums over compositions}, where we look at composition sums over each part of a composition

Some new identities on the Apostol-Bernoulli polynomials of higher order derived from Bernoulli basis

In the present paper, we obtain new interesting relations and identities of the Apostol-Bernoulli polynomials of higher order, which are derived using a Bernoulli polynomial basis. Finally, by utilizing our method, we also derive formulas for the convolutions of Bernoulli and Euler polynomials, expressed via Apostol-Bernoulli polynomials of higher order.Comment: 8 pages, submitte

On the computation of classical, boolean and free cumulants

This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cumulants. This is realized by using a suitable polynomial representation of Abel polynomials. The algorithm relies on the classical umbral calculus, a symbolic language introduced by Rota and Taylor in 1994, that is particularly suited to be implemented by using software for symbolic computations. Here we give a MAPLE procedure. Comparisons with existing procedures, especially for conversions between moments and free cumulants, as well as examples of applications to some well-known distributions (classical and free) end the paper.Comment: 14 pages. in press, Applied Mathematics and Computatio