Ana F. Loureiro

Iteration functions for pth roots of complex numbers

A novel way of generating higher-order iteration functions for the computation of pth roots of complex numbers is the main contribution of the present work. The behavior of some of these iteration functions will be analyzed and the conditions on the starting values that guarantee the convergence will be stated. The illustration of the basins of attractions of the pth roots will be carried out by some computer generated plots. In order to compare the performance of the iterations some numerical examples will be considered.

On a polynomial sequence associated with the Bessel operator

By means of the Bessel operator a polynomial sequence is constructed to which several properties are given. Among them are its explicit expression, the connection with the Euler numbers, and its integral representation via the Kontorovich-Lebedev transform. Despite its non-orthogonality (with respect to an

Central factorials under the Kontorovich–Lebedev transform of polynomials

L_{2}

On the convergence of Schröder iteration functions for pth roots of complex numbers

-inner product), it is possible to associate to the canonical element of its dual sequence a positive-definite measure as long as certain stronger constraints are imposed.

Unique positive solution for an alternative discrete Painlevé I equation

In this paper, we show that slight modifications of the Kontorovich–Lebedev (KL) transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.

On Especial Cases of Boas-Buck-Type Polynomial Sequences

In this work a condition on the starting values that guarantees the convergence of the Schroder iteration functions of any order to a pth root of a complex number is given. Convergence results are derived from the properties of the Taylor series coefficients of a certain function. The theory is illustrated by some computer generated plots of the basins of attraction.

Polynomial sequences associated with the classical linear functionals

We show that the alternative discrete Painlevé I equation (alt-) has a unique solution which remains positive for all . Furthermore, we identify this positive solution in terms of a special solution of the second Painlevé equation () involving the Airy function . The special-function solutions of involving only the Airy function therefore have the property that they remain positive for all and all , which is a new characterization of these special solutions of and alt-.

Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

After a slight modification, the Kontorovich-Lebedev transform is an automorphism in the vector space of polynomials. The action of this transformation over special cases of Boas-Buck-type polynomial sequences is under analysis.

Quadratic decomposition of Appell sequences

This work in mainly devoted to the study of polynomial sequences, not necessarily orthogonal, defined by integral powers of certain first order differential operators in deep connection to the classical polynomials of Hermite, Laguerre, Bessel and Jacobi. This connection is streamed from the canonical element of their dual sequences. Meanwhile new Rodrigues-type formulas for the Hermite and Bessel polynomials are achieved.

New results on the Bochner condition about classical orthogonal polynomials

By [arXiv:1604.00528], a list of possible holonomy algebras for pseudo-Riemannian manifolds with an indecomposable torsion free