Zeros of Gegenbauer and Hermite polynomials and connection coefficients

In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.

     

Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials

We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials p_k(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑ_m(x) p_n(x;q), where ϑ_m(x) is x^mor (x;q)_m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail. This revised version was published online in June 2006 with corrections to the Cover Date.     

Another connection between orthogonal polynomials and L-orthogonal polynomials

We consider a connection that exists between orthogonal polynomials associated with positive measures on the real line and orthogonal Laurent polynomials associated with strong measures of the class S³[0,β,b]. Examples are given to illustrate the main contribution in this paper.     

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.     

On connection coefficients of some perturbed of arbitrary order of the Chebyshev polynomials of second kind

Orthogonal polynomials satisfy a recurrence relation of order two defined by two sequences of coefficients. If we modify one of these recurrence coefficients at a certain order, we obtain the so-called perturbed orthogonal sequence. In this work, we analyse perturbed Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow us to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection coefficients obtained, we derive some results about zeros at the origin. The analysis is valid for arbitrary order of perturbation.     

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.     

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.     

Menke points on the real line and their connection to classical orthogonal polynomials

We investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1].     

Limit relations between q-Krall type orthogonal polynomials

In this paper, we consider a natural extension of several results related to Krall-type polynomials introducing a modification of a q-classical linear functional via the addition of one or two mass points. The limit relations between the q-Krall type modification of big q-Jacobi, little q-Jacobi, big q-Laguerre, and other families of the q-Hahn tableau are established.     

Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle

Rakhmanov's theorem for orthogonal polynomials on the unit circle gives a sufficient condition on the orthogonality measure for orthogonal polynomials on the unit circle, in order that the reflection coefficients (the recurrence coefficients in the Szegő recurrence relation) converge to zero. In this paper we give the analog for orthogonal matrix polynomials on the unit circle.     

On linearization and connection coefficients for generalized Hermite polynomials

We consider the problem of finding explicit formulas, recurrence relations and sign properties for both connection and linearization coefficients for generalized Hermite polynomials. Most of the computations are carried out by the computer algebra system Maple using appropriate algorithms.     

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.     

Unification of Stieltjes‐Calogero type relations for the zeros of classical orthogonal polynomials

The classical orthogonal polynomials (COPs) satisfy a second‐order differential equation of the form σ(x)y^′′+τ(x)y^′+λy = 0, which is called the equation of hypergeometric type (EHT). It is shown that two numerical methods provide equivalent schemes for the discrete representation of the EHT. Thus, they lead to the same matrix eigenvalue problem. In both cases, explicit closed‐form expressions for the matrix elements have been derived in terms only of the zeros of the COPs. On using the equality of the entries of the resulting matrices in the two discretizations, unified identities related to the zeros of the COPs are then introduced. Hence, most of the formulas in the literature known for the roots of Hermite, Laguerre and Jacobi polynomials are recovered as the particular cases of our more general and unified relationships. Furthermore, we present some novel results that were not reported previously. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series

We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p-adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p-adic type.     

Special linear combinations of orthogonal polynomials

The paper lists a number of problems that motivate consideration of special linear combinations of polynomials, orthogonal with the weight p(x) on the interval (a,b). We study properties of the polynomials, as well as the necessary and sufficient conditions for their orthogonality. The special linear combinations of Chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations.     

On q-orthogonal polynomials over a collection of complex origin intervals related to little q-Jacobi polynomials

We construct the sequence of orthogonal polynomials with respect to an inner product which is defined by q-integrals over a collection of intervals in the complex plane. We prove that they are connected with little q-Jacobi polynomials. For such polynomials we discuss a few representations, a recurrence relation, a difference equation, a Rodrigues-type formula and a generating function. 2000 Mathematics Subject Classification Primary—33D45, 42C05