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The adjunction property, recently introduced for Sheffer polynomial sets, is considered in the case of Appell polynomials. The particular case of adjoint Appell-Euler and Appell-Bernoulli polynomials of the first kind is analyzed. [ABSTRACT FROM AUTHOR]

The internal and external Robin problems for the Laplace equation in bounded starlike domains are addressed. We show how to derive the relevant solutions by using a suitable Fourier series-like method. Numerical results are specifically obtained considering three-dimensional domains whose boundary is defined by a generalization of the so-called "superformula" introduced by Gielis. By using the computer algebra code Mathematica©, truncated series approximations of the solutions are determined. Our findings are in good agreement with the theoretical results on the Fourier series due to Carleson. [ABSTRACT FROM AUTHOR]

By using the classical Hausdorff identity it is possible to define general families of Gould-Hopper and Laguerre polynomials. In this article we obtain further generalizations of Laguerre polynomials, and construct the relevant properties by exploiting a standard technique associated with the monomiality principle. [ABSTRACT FROM AUTHOR]

The Rayleigh–Ritz and the inverse iteration methods are used in order to compute the eigenvalues of Fredholm–Stieltjes integral equations, i.e. Fredholm equations with respect to suitable Stieltjes-type measures. Some applications to the so-called 'charged' (in German 'belastete') integral equation, and particularly the problem of computing the eigenvalues of a string charged by a finite number of cursors are given. [ABSTRACT FROM AUTHOR]

We use the inverse iteration method in order to approximate the first eigenvalues and eigenfunctions of Fredholm-Stieltjes integral equations of second kind, i.e. Fredholm equations with respect to suitable Stieltjes-type measures. Some applications to the problem of a string with varying density ρ(x) and charged by a finite number of cursors are given. [ABSTRACT FROM AUTHOR]

We introduce suitable exponential operators and use the multi-dimensional polynomials considered by Hermite, and subsequently studied by P. Appell and J. Kampé de Fériet, H.W. Gould--A.T. Hopper, G. Dattoli et al. in order to obtain explicit solutions of classical and generalized PDE problems in the half-plane y > 0 , connecting systems with equations, and including some problems with degeneration on the x-axis. [ABSTRACT FROM AUTHOR]

We use the multi-dimensional polynomials considered by Hermite, and subsequently studied by P. Appell and J. Kampé de Fériet, in order to obtain explicit solutions of pseudo-classical PDE problems in the half-plane y > 0. We consider systems of PDE, including some problems with degeneration on the x-axis. [ABSTRACT FROM AUTHOR]

Let {Pn(x)∞n=0} be a sequence of polynomials of degree n. We define two sequences of differential operators φn and ψn satisfying the following properties Φn(Pn(x)) = Pn-1(x), Ψn(Pn(x)) = Pn+1(x). By constructing these two operators for some classes of special functions, we determine their differential equations via the factorization method introduced in [3]. We illustrate our method by including classical orthogonal polynomials, d-orthogonal polynomials, confluent hypergeometric functions and hypergeometric functions as the applications. [ABSTRACT FROM AUTHOR]

The monomiality principle has allowed the derivation, in a fairly straightforward way, of the properties of many families of special polynomials, recognized as quasi monomials. The family of quasimonomials is large, but not all the special polynomials can be framed within this category. In this paper we will prove the existence of a larger family, including the quasi-monomials, by taking advantage from the umbral calculus and by introducing the concept of monumbrality, which appears very useful to get a deeper understanding in the nature and role of special polynomials. [ABSTRACT FROM AUTHOR]

We use an operational method to show that Legendre polynomials can be viewed as discrete convolutions of Laguerre polynomials. It is furthermore shown that they can be derived as the particular case of a new family of two-variable orthogonal polynomials, whose properties are studied with some detail. It is furthermore shown that this point of view allows the derivation of new properties of Legendre polynomials and it is also shown how it can be extended to the Jacobi family [ABSTRACT FROM AUTHOR]