Orthogonal polynomials on the unit circle via a polynomial mapping on the real line

Let {[Phi]n}n[greater-or-equal, slanted]0 be a sequence of monic orthogonal polynomials on the unit circle (OPUC) with respect to a symmetric and finite positive Borel measure d[mu] on [0,2[pi]] and let -1,[alpha]0,[alpha]1,[alpha]2,... be the associated sequence of Verblunsky coefficients. In this paper we study the sequence of monic OPUC whose sequence of Verblunsky coefficients iswhere b1,b2,...,bN-1 are N-1 fixed real numbers such that bj[set membership, variant](-1,1) for all j=1,2,...,N-1, so that is also orthogonal with respect to a symmetric and finite positive Borel measure on the unit circle. We show that the sequences of monic orthogonal polynomials on the real line (OPRL) corresponding to {[Phi]n}n[greater-or-equal, slanted]0 and (by Szegö's transformation) are related by some polynomial mapping, giving rise to a one-to-one correspondence between the monic OPUC on the unit circle and a pair of monic OPRL on (a subset of) the interval [-1,1]. In particular we prove thatd[mu][small tilde]([theta])=[zeta]N-1([theta])sin[theta]sin[theta]N([theta])d[mu]([theta]N([theta]))[theta]N'([theta]),supported on (a subset of) the union of 2N intervals contained in [0,2[pi]] such that any two of these intervals have at most one common point, and where, up to an affine change in the variable, [zeta]N-1 and cos[theta]N are algebraic polynomials in cos[theta] of degrees N-1 and N (respectively) defined only in terms of [alpha]0,b1,...,bN-1. This measure induces a measure on the unit circle supported on the union of 2N arcs, pairwise symmetric with respect to the real axis. The restriction to symmetric measures (or real Verblunsky coefficients) is needed in order that Szegö's transformation may be applicable.http://www.sciencedirect.com/science/article/B6TYH-4NNWCG5-1/1/5cc167c4a58817de62d99d3dd5c88e3

On the Krall-type discrete polynomials

In this paper we present a unified theory for studying the so-called Krall-type discrete orthogonal polynomials. In particular, the three-term recurrence relation, lowering and raising operators as well as the second order linear difference equation that the sequences of monic orthogonal polynomials satisfy are established. Some relevant examples of q-Krall polynomials are considered in detail.http://www.sciencedirect.com/science/article/B6WK2-4CC2YF9-1/1/1bbcf94cc1184e679b497c3b8e754b2

Explicit inverses of some tridiagonal matrices

We give explicit inverses of tridiagonal 2-Toeplitz and 3-Toeplitz matrices which generalize some well-known results concerning the inverse of a tridiagonal Toeplitz matrix.http://www.sciencedirect.com/science/article/B6V0R-42KDHCJ-2/1/d6207c3bf9c66184210a9296f72fe1e

Explicit inverses of some tridiagonal matrices

We give explicit inverses of tridiagonal 2-Toeplitz and 3-Toeplitz matrices which generalize some well-known results concerning the inverse of a tridiagonal Toeplitz matrix.http://www.sciencedirect.com/science/article/B6V0R-42KDHCJ-2/1/d6207c3bf9c66184210a9296f72fe1e

WKB Approximation and Krall-Type Orthogonal Polynomials

We give a unified approach to the Krall-type polynomials orthogonal withrespect to a positive measure consisting of an absolutely continuous one‘perturbed’ by the addition of one or more Dirac deltafunctions. Some examples studied by different authors are considered from aunique point of view. Also some properties of the Krall-type polynomials arestudied. The three-term recurrence relation is calculated explicitly, aswell as some asymptotic formulas. With special emphasis will be consideredthe second order differential equations that such polynomials satisfy. Theyallow us to obtain the central moments and the WKB approximation of thedistribution of zeros. Some examples coming from quadratic polynomialmappings and tridiagonal periodic matrices are also studied

On linearly related sequences of derivatives of orthogonal polynomials

We discuss an inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families (Pn)n and (Qn)n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as for all n=0,1,2,..., where M and N are fixed nonnegative integer numbers, and ri,n and si,n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with (Pn)n and (Qn)n (resp.). Assuming 0[less-than-or-equals, slant]m[less-than-or-equals, slant]k, we prove the existence of four polynomials [Phi]M+m+i and [Psi]N+k+i, of degrees M+m+i and N+k+i (resp.), such that the (k-m)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If k=m, then u and v are connected by a rational modification. If k=m+1, then both u and v are semiclassical linear functionals, which are also connected by a rational modification. When k>m, the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary differential equation of order k-m with polynomial coefficients.http://www.sciencedirect.com/science/article/B6WK2-4SSNDCC-2/1/2844d686d4f273e5ddf7b4d7146c9ee