We reconsider some families of orthogonal polynomials, within the framework of the so called monomiality principle. We show that the associated operational formalism allows the framing of the polynomial orthogonality using an algebraic point of view. Within such a framework, we introduce families of pseudo-orthogonal polynomials, namely polynomials, not orthogonal under the ordinary definition,...

Inspired by Wilmshurst’s conjecture, W. Li and A. Wei (2009) initiated a probabilistic study concerning the zeros of harmonic polynomials. They derived a Kac-Rice type formula for the expected number of zeros of random harmonic polynomials p(z) + q(z) with independent Gaussian coefficients. They also provided asymptotics in the case when p and q are sampled independently from the the (complex) ...

In this paper, a common generalization of the Rogers-Ramanujan series and the generating function for basis partitions is studied. This leads naturally to a sequence of polynomials, called BsP-polynomials. In turn, the BsP-polynomials provide simultaneously a proof of the Rogers-Ramanujan identities and a new, more rapidly converging series expansion for the basis partition generating function....

The connection between the Bessel discrete variable basis expansion and a specific form of an orthogonal set of Jacobi polynomials is demonstrated. These so-called Zernike polynomials provide alternative series expansions of suitable functions over the unit interval. Expressing a Bessel function in a Zernike expansion provides a straightforward method of generating series identities. Furthermor...

We present a method for solving the two-dimensional equation of transfer. The method can be extended easily to the general linear transport problem. The used technique allows us to reduce the two-dimensional equation to a system of one-dimensional equations. The idea of using the spectral method for searching for solutions to the multidimensional transport problems leads us to a solution for al...

Numerical simulations of plasma heating with waves in the ion cyclotron range of frequencies (ICRF) require to iteratively couple a solver for wave propagation in plasmas with a solver of the quasilinear kinetic equation. Among the codes developed for this purpose, the TORIC-SSFPQL package is characterized by its high execution speed. The kinetic code SSFPQL, however, was based on a somewhat si...

An attack on a recently proposed authentication scheme of Shpilrain and Ushakov is presented. The public information allows the derivation of a system of polynomial equations for the secret key bits. Our attack uses simple elimination techniques to distill linear equations. For the proposed parameter choice, the attack often finds secret keys or alternative secret keys within minutes with moder...