Asymptotic properties of Laguerre–Sobolev type orthogonal polynomials

In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product

Extensions of Szegö's theory of orthogonal polynomials,II

Let {? _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior of? _n (dμ) when logμ'∈L ¹. In what follows we will discuss the asymptotic behavior of the ratio φ_n(dμ ₁)/φ_n(dμ ₂) off the unit circle in casedμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=g dμ ₁ whereg≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0, for the monic polynomials Φ _n . The consequences for orthogonal polynomials on the real line are also discussed.     

Extensions of Szegö's theory of orthogonal polynomials,III

Let {ø _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofø _n (dμ) when logμ′∈L ¹. In what follows we will discuss the asymptotic behavior of the ratioø _n (dμ ₂)/ø _n (dμ ₁) on the unit circle whendμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=gdμ ₁, where g≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0 for the monic polynomial Φ _n . The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.     

Discrete orthogonal polynomials – polynomial modification of a classical functional

Polynomial modifications of a classical discrete linear functional are examined in detail, in particular when the new linear functional remains classical. New addition formulas are deduced for Charlier, Meixner and Hahn polynomials from the Christoffei representation and results are also given for a particular generalized Meixner family.     

A new class of three-variable orthogonal polynomials and their recurrences relations

A new class of three-variable orthogonai polynomials,defined as eigenfunctions of a second order PDE operator,is studied.These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron,and can be taken as an extension of the 2-D Steiner domain.The polynomials can be viewed as Jacobi polynomials on such a domain.Three- term relations are derived explicitly.The number of the individual terms,involved in the recurrences relations,are shown to be independent on the total degree of the polynomials.The numbers now are determined to be five and seven,with respect to two conjugate variables z,(?) and a real variable r, respectively.Three examples are discussed in details,which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds,and Legendre polynomials.     

A note on the zeros of Freud–Sobolev orthogonal polynomials

We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e^-x4${\mathrm{e}}^{-x^4}$ on R$\mathbb{R}$ are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e^-x4${\mathrm{e}}^{-x^4}$. Some numerical examples are shown.     

Construction and implementation of asymptotic expansions for Jacobi–type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to \(\infty \). These are defined on the interval [?1, 1] with weight function

$$w(x)=(1-x)^{\alpha}(1+x)^{\beta}h(x), \quad \alpha,\beta>-1 $$

and h(x) a real, analytic and strictly positive function on [?1, 1]. This information is available in the work of Kuijlaars et al. (Adv. Math. 188, 337–398 2004), where the authors use the Riemann–Hilbert formulation and the Deift–Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in \(\mathcal {O}(n)\) operations, rather than \(\mathcal {O}(n^{2})\) based on the recurrence relation.

     

q-Discrete Painlevé equations for recurrence coefficients of modified q-Freud orthogonal polynomials

We present an asymmetric q-Painlevé equation. We will derive this using q-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this q-Painlevé equation (up to a simple transformation). We will show a stable method of computing a special solution, which gives the recurrence coefficients. We establish a connection with α-q-P_V.     

Szegő’s theorem for matrix orthogonal polynomials

We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szeg?’s theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager.     

Laguerre–Freud equations for three families of hypergeometric discrete orthogonal polynomials

The Cholesky factorization of the moment matrix is considered for discrete orthogonal polynomials of hypergeometric type. We derive the Laguerre–Freud equations when the first moments of the weights are given by the ₁F₂, ₂F₂, and ₃F₂ generalized hypergeometric series.     

Matrix orthogonal Laurent polynomials and two-point Padé approximants

This paper is concerned with double sequencesC={C _n} _n ^=–/ of Hermitian matrices with complex entriesC _n M _s×s ) and formal Laurent seriesL ₀(z)=– _k=1 C _–k z ^k andL (z)= _k=0 C _k z ^–k . Making use of a Favard-type theorem for certain sequences of matrix Laurent polynomials which was obtained previously in [1] we can establish the relation between the matrix counterpart of the so-calledT-fractions and matrix orthogonal Laurent polynomials. The connection with two-point Padé approximants to the pair (L ₀,L ) is also exhibited proving that such approximants are Hermitian too. Finally, error formulas are also given.     

Zeros of para–orthogonal polynomials and linear spectral transformations on the unit circle

We study the interlacing properties of zeros of para–orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle dµ and para–orthogonal polynomials associated with a modification of dµ by the addition of a pure mass point, also called Uvarov transformation. Moreover, as a direct consequence of our approach, we present some results related with the Christoffel transformation.