Variable-precision recurrence coefficients for nonstandard orthogonal polynomials

Inequalities recently conjectured for all zeros of Jacobi polynomials of all degrees n are modified and conjectured to hold (in reverse direction) in considerably larger domains of the (α,β)-plane.      

Orthogonality measures for orthogonal matrix polynomials with periodic coefficients of recurrence relations

Mathematical Notes -     

Asymptotics for orthogonal polynomials defined by a recurrence relation

Asymptotic expansions are given for orthogonal polynomials when the coefficients in the three-term recursion formula generating the orthogonal polynomials form sequences of bounded variation.     

Differential coefficients of orthogonal matrix polynomials

We find explicit formulas for raising and lowering first order differential operators for orthogonal matrix polynomials. We derive recurrence relations for the coefficients in the raising and lowering operators. Some examples are given.     

Formulas for recurrence coefficients of orthogonal polynomials related to Lorentzian-like weights

One considers the recurrence relation of orthogonal polynomials related to weights |t|^A(1+t^2r/c^2r)^-B on the whole real line, for various integer exponents 2r, and real A>-1, B>0.     

Spectral measures corresponding to orthogonal polynomials with unbounded recurrence coefficients

Sufficient conditions are presented for the existence of an absolutely continuous part for the measure of orthogonality for a system of polynomials with unbounded recurrence coefficients. Results are obtained by analyzing the spectral measure of a related self-adjoint operator.Communicated by Paul Nevai.     

Rodrigues formula and recurrence coefficients for non-symmetric Dunkl-classical orthogonal polynomials

The Ramanujan Journal - In this paper, we establish a distributional Rodrigues formula for non-symmetric Dunkl-classical orthogonal polynomial sequences. Then, we use this formula to determine,...     

Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

We consider orthogonal polynomials , where n is the degree of the polynomial and N is a discrete parameter. These polynomials are orthogonal with respect to a varying weight W_N which depends on the parameter N and they satisfy a recurrence relation with varying recurrence coefficients which we assume to be varying monotonically as N tends to infinity. We establish the existence of the limit and link this limit to an external field for an equilibrium problem in logarithmic potential theory.     

Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials

We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations.     

On the limit behavior of recurrence coefficients for multiple orthogonal polynomials

In this paper we investigate general properties of the coefficients in the recurrence relation satisfied by multiple orthogonal polynomials. The results include as particular cases Angelesco and Nikishin systems.     

On differential properties for bivariate orthogonal polynomials

In this paper, we consider bivariate orthogonal polynomials associated with a quasi-definite moment functional which satisfies a Pearson-type partial differential equation. For these polynomials differential properties are obtained. In particular, we deduce some structure and orthogonality relations for the successive partial derivatives of the polynomials.      

Absolutely continuous measures for systems of orthogonal polynomials with unbounded recurrence coefficients

This paper considers systems of orthogonal polynomials which satisfy a three-term recurrence relation in which the recursion coefficients are unbounded. Conditions are imposed on the coefficient sequences to establish that the corresponding measure of orthogonality is absolutely continuous.     

A companion matrix analogue for orthogonal polynomials

Given a set of orthogonal polynomials {P_i(x)}, it is shown that associated with a polynomial a(x)=∑a_ip_i(x) there is a matrix A which possesses several of the properties of the usual companion form matrix C. An alternative and possibly preferable form A' is also suggested. A similarity transformation between A [orA'] and C is given. If b(x) is another polynomial then the matrix b(A) [or b(A')] has properties like those of b(C), relating to the greatest common divisor of a(x) and b(x).     

Connection coefficients for Laguerre-Sobolev orthogonal polynomials

Laguerre-Sobolev polynomials are orthogonal with respect to an inner product of the form , where α>−1, λ?0, and , the linear space of polynomials with real coefficients. If dμ(x)=xαe−xdx, the corresponding sequence of monic orthogonal polynomials {Q_n^(α,λ)(x)} has been studied by Marcellán et al. (J. Comput. Appl. Math. 71 (1996) 245-265), while if dμ(x)=δ(x)dx the sequence of monic orthogonal polynomials {L_n^(α)(x;λ)} was introduced by Koekoek and Meijer (SIAM J. Math. Anal. 24 (1993) 768-782). For each of these two families of Laguerre-Sobolev polynomials, here we give the explicit expression of the connection coefficients in their expansion as a series of standard Laguerre polynomials. The inverse connection problem of expanding Laguerre polynomials in series of Laguerre-Sobolev polynomials, and the connection problem relating two families of Laguerre-Sobolev polynomials with different parameters, are also considered.     

Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation

Transformations of the measure of orthogonality for orthogonal polynomials, namely Freud transformations, are considered. Jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations. Dynamics of the coefficients are described by generalized Toda equations. The classical Toda lattice equations are the simplest special case of dynamics of the coefficients under the Freud transformation of the measure of orthogonality. Also dynamics of Hankel determinants, its minors and other notions corresponding to the orthogonal polynomials are studied.     

The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights

We consider orthogonal polynomials on the real line with respect to a weight and in particular the asymptotic behaviour of the coefficients a_n,N and b_n,N in the three-term recurrence xπ_n,N(x)=π_n+1,N(x)+b_n,Nπ_n,N(x)+a_n,Nπ_n−1,N(x). For one-cut regular V we show, using the Deift-Zhou method of steepest descent for Riemann-Hilbert problems, that the diagonal recurrence coefficients a_n,n and b_n,n have asymptotic expansions as n→∞ in powers of 1/n² and powers of 1/n, respectively.