A refinement of a conjecture of Gasper concerning the values of (α, β), −1/2 < β < 0, −1 < α + β < 0, for which the inequalities

The multivariate orthogonal polynomials are related to a family of operators whose matrix representations are block Jacobi matrices. A sufficient condition is given so that these operators, in general unbounded, are commuting and selfadjoint. The spectral theorem for these operators is used to establish the existence of the measure of orthogonality in Favard's theorem.

The operator Lμ : f 7→ ∫ f(x)−f(y) x−y dμ(y) is, for a compactly supported measure μ with an L density, a closed, densely defined operator on L(μ). We show that the operator Q = pLμ−qLμ has polynomial eigenfunctions if and only if μ is a free Meixner distribution. The only time Q has orthogonal polynomial eigenfunctions is if μ is a semicircular distribution. More generally, the only time the o...

Given {Pn}n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e., Qn(x) = Pn(x) + a1Pn−1(x) + · · ·+ akPn−k, ak 6= 0, n > k. Necessary and sufficient conditions are given for the orthogonality of the sequence {Qn}n≥0. An interesting interpretation in terms of the Jacobi matrices associated with {Pn}n≥0 and {Qn}n≥0 i...

Angelesco systems of measures with Jacobi type weights are considered. For such systems, strong asymptotics for the related multiple orthogonal polynomials are found as well as the Szegő-type functions. In the procedure, an approach from Riemann-Hilbert problem plays a fundamental role.

In our paper we construct a polynomial Schauder basis (pα,β,n)n∈N0 of optimal degree with Jacobi orthogonality. A candidate for such a basis is given by the use of some wavelet theoretical methods, which were already successful in case of Tchebysheff and Legendre orthogonality. To prove that this sequence is in fact a Schauder basis for C[−1, 1] and as the main difficulty of the whole proof we ...

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relation...

Denote by xn,k(α, β), k = 1, . . . , n, the zeros of the Jacobi polynomial P (α,β) n (x). It is well known that xn,k(α, β) are increasing functions of β and decreasing functions of α. In this paper we investigate the question of how fast the functions 1 − xn,k(α, β) decrease as β increases. We prove that the products tn,k(α, β) := fn(α, β) (1− xn,k(α, β)), where fn(α, β) = 2n2 + 2n(α + β + 1) +...

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points ...

We survey the analytic theory of matrix orthogonal polynomials. MSC: 42C05, 47B36, 30C10 keywords: orthogonal polynomials, matrix-valued measures, block Jacobi matrices, block CMV matrices