Quadratic Decomposition of Bivariate Orthogonal Polynomials

Abstract

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of Bäcklund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.

The Properties of Topological Manifolds of Simplicial Polynomials

“…(3) Is it possible to relax the requirement of positivity of polynomials in the formation of simplicial polynomials as a generalization? And, finally, (4) what are the properties of the resulting topological manifolds generated by them? This article addresses these questions in relative detail by combining the elements of geometric topology and algebraic geometry.…”

“…Often, monomials play important roles in determining the decomposability of polynomials. Interestingly, special classes of polynomials, for example, Hermite polynomials, can reveal quadratic decomposition in bivariate forms under certain conditions [4]. First, we present the preliminary concepts, motivations and the contributions made in this paper in the following sections.…”

Hypergeometric expressions for type I Jacobi–Piñeiro orthogonal polynomials with arbitrary number of weights

“…Recently, the importance of multiple orthogonal polynomials in the Favard spectral description of banded bounded semi-infinite matrices has come to light. This connection has been explored in various sources such as [9,14,15], see also [13]. Additionally, their significance in the context of Markov chains and random walks beyond birth and death has been established, as seen in [7,[10][11][12].…”

Untitled

“…In our work presented in [6], we offer explicit hypergeometric expressions for all type I polynomials within the realm of multiple Hahn orthogonal polynomials under the Askey scheme [5,18], a context encompassing only two weights. Through these efforts, we have successfully constructed bidiagonal factorizations and characterized positive ones [8], as elaborated in [9,14].…”