We derive and factorize the fourth-order difference equations satisfied by orthogonal polynomials obtained from some modifications of the recurrence coefficients of classical discrete orthogonal polynomials such as: the associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independe...

orthogonal zero interpolants (ozi) are polynomials which interpolate the “zero-function” at a finite number of pre-assigned nodes and satisfy orthogonality condition. ozi’s can be constructed by the 3-term recurrence relation. these interpolants are found useful in the solution of constrained approximation problems and in the structure of gauss-type quadrature rules. we present some theoretical...

Let ν be either ω ∈ C \ {0} or q ∈ C \ {0, 1}, and let Dν be the corresponding difference operator defined in the usual way either by Dωp(x) = p(x+ω)−p(x) ω or Dqp(x) = p(qx)−p(x) (q−1)x . Let U and V be two moment regular linear functionals and let {Pn(x)}n≥0 and {Qn(x)}n≥0 be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete ort...

It is shown that a conjecture of E.A. Rakhmanov is true concerning the zero distribution of orthogonal polynomials with respect to a measure having a discrete real support. We also discuss the case of extremal polynomials with respect to some discrete L p {norm, 0 < p 1, and give an extension to complex supports. Furthermore, we present properties of weighted Fekete points with respect to discr...

Szeg˝ o's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [−1, 1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials Λ, and leads to a new orthogonality structure in the module Λ × Λ. This structure can be interpreted in terms of a 2 × 2 matrix measure on [−1, 1], a...

We study a surprising phenomenon related to the representation of a cloud of data points using polynomials. We start with the previously unnoticed empirical observation that, given a collection (a cloud) of data points, the sublevel sets of a certain distinguished polynomial capture the shape of the cloud very accurately. This distinguished polynomial is a sum-of-squares (SOS) derived in a simp...

We consider Koornwinder’s method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with ...

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures, and to quantitative results on the expected number of zeros in various sets. In the simplest case of Kac polynomials, given by the linear combinations of monomials with i.i.d. random coefficients, it is well known that th...