Polynomials orthogonal with respect to a perturbation of the Freud weight function by the addition of a mass point at zero are considered. These polynomials, called Freud-type orthogonal polynomials, satisfy a second order linear differential equation with varying polynomial coefficients. It plays an important role in the electrostatic interpretation for the distribution of zeros of the corresp...

The orthogonal polynomials pn satisfy Turán’s inequality if p 2 n(x)− pn−1(x)pn+1(x) ≥ 0 for n ≥ 1 and for all x in the interval of orthogonality. We give general criteria for orthogonal polynomials to satisfy Turán’s inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the q–ultraspherical polynomials.

We rst give a brief survey of some aspects of orthogonal polynomials. The three-term recurrence relation gives a tridiagonal matrix and the corresponding Ja-cobi operator gives useful information about the orthogonalizing measure and the asymptotic behavior of the zeros of the orthogonal polynomials. The Toda lattice and other similar dynamical systems (Langmuir lattice or Kac-Van Moerbeke latt...

In this paper, using both an analytic and algorithmic approach, we derive the coefficients Dm(n, a) of the multiplication formula pn(ax) = n ∑ m=0 Dm(n, a)pm(x) or the translation formula pn(x +a) = n ∑ m=0 Dm(n, a)pm(x), where {pn}n≥0 is an orthogonal polynomial set, including the classical continuous orthogonal polynomials, the classical discrete orthogonal polynomials, the q-classical orthog...

In this paper multiple orthogonal polynomials defined using orthogonality conditions spread out over r different measures are considered. We study multiple orthogonal polynomials on the real line, as well as on the semicircle (complex polynomials orthogonal with respect to the complex-valued inner products (f, g)k = ∫ π 0 f(e)g(e)wk(e ) dθ, for k = 1, 2, . . . , r). For r = 1, in the real case ...

We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials.

Constrained orthogonal polynomials have been recently introduced in the study of the HohenbergKohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean scalar product, to a given i-dimensional subspace Ei of polyno...

Constrained orthogonal polynomials have been recently introduced in the study of the HohenbergKohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean inner product, to a given i-dimensional subspace Ei of polynom...

R |xα| dμ(x) < ∞ (α ∈ (Z≥0)) and the support of μ has nonempty interior. Let Pn consist of all polynomials p of degree ≤ n such that ∫ Rd pq dμ = 0 for all polynomials q of degree < n. Then Pn has the same dimension ( n+d−1 n ) as the space of homogeneous polynomials of degree n in d variables. Furthermore, the spaces Pn (n = 0, 1, 2, . . .) are mutually orthogonal in L(μ). We call {Pn}n=0 a sy...

In this paper two types of multiple orthogonal polynomials on the semicircle with respect to a set of r different weight functions are defined. Such polynomials are generalizations of polynomials orthogonal on the semicircle with respect to a complex-valued inner product 1⁄2f ; g 1⁄4 R p 0 f e ih g eih w eih dh. The existence and uniqueness of introduced multiple orthogonal polynomials for cert...