Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials

The asymptotic contracted measure of zeros of a large class of orthogonal polynomials is explicitly given in the form of a Lauricella function. The polynomials are defined by means of a three-term recurrence relation whose coefficients may be unbounded but vary regularly and have a different behaviour for even and odd indices. Subclasses of systems of orthogonal polynomials having their contracted measure of zeros of regular, uniform, Wigner, Weyl, Karamata and hypergeometric types are explicitly identified. Some illustrative examples are given.     

Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures

Let μ be a finite positive Borel measure with compact support consisting of an interval plus a set of isolated points in , such that μ^′>0 almost everywhere on [c,d]. Let , be a sequence of polynomials, , with real coefficients whose zeros lie outside the smallest interval containing the support of μ. We prove ratio and relative asymptotics of sequences of orthogonal polynomials with respect to varying measures of the form dμ/w_2n. In particular, we obtain an analogue for varying measures of Denisov's extension of Rakhmanov's theorem on ratio asymptotics. These results on varying measures are applied to obtain ratio asymptotics for orthogonal polynomials with respect to fixed measures on the unit circle and for multi-orthogonal polynomials in which the measures involved are of the type described above.     

Zeros of orthogonal polynomials on the real line

Let p_n(x) be the orthonormal polynomials associated to a measure dμ of compact support in . If Esupp(dμ), we show there is a δ>0 so that for all n, either p_n or p_n+1 has no zeros in (E−δ,E+δ). If E is an isolated point of supp(μ), we show there is a δ so that for all n, either p_n or p_n+1 has at most one zero in (E−δ,E+δ). We provide an example where the zeros of p_n are dense in a gap of supp(dμ).     

Asymptotics of derivatives of orthogonal polynomials on the real line

We show that uniform asymptotics of orthogonal polynomials on the real line imply uniform asymptotics for all their derivatives. This is more technically challenging than the corresponding problem on the unit circle. We also examine asymptotics in the L ₂ norm. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.     

Q-operator and factorised separation chain for Jack polynomials

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P_λ^(1/g) (χ₁, …, χ_n) …, χ_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_zk we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P_λ^(1/g) (x₁, …, x_k, 1, … 1). Using the operator Q_z for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.     

An update on orthogonal polynomials and weighted approximation on the real line

We briefly review the state of orthogonal polynomials on (–, ), concentrating on analytic aspects, such as asymptotics and bounds on orthogonal polynomials, their zeros and their recurrence coefficients. We emphasize results rather than proofs. We also discuss applications to mean convergence of orthogonal expansions, Lagrange interpolation, Jackson-Bernstein theorems and the weighted incomplete polynomial approximation problem.     

Ratio asymptotic of Hermite-Padé orthogonal polynomials for Nikishin systems. II

We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures N(σ₁,…,σm) such that for each k, σk has constant sign on its support consisting on an interval , on which almost everywhere, and a set without accumulation points in .     

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1,1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann–Hilbert (RH) problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the RH problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order 1/n terms in the expansions. A critical step in the analysis of the RH problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order. In addition, the RH approach gives us also strong asymptotics of the orthogonal polynomials near the algebraic singularities in terms of Bessel functions.     

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].     

Ratio asymptotics for orthogonal rational functions on an interval

Let {α₁,α₂,…} be a sequence of real numbers outside the interval [−1,1] and μ a positive bounded Borel measure on this interval satisfying the Erd s–Turán condition μ′>0 a.e., where μ′ is the Radon–Nikodym derivative of the measure μ with respect to the Lebesgue measure. We introduce rational functions _n(x) with poles {α₁,…,α_n} orthogonal on [−1,1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of _n+1(x)/_n(x) as n tends to infinity under certain assumptions on the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation satisfied by the orthonormal functions.     

Strong asymptotics for Sobolev orthogonal polynomials in the complex plane

We obtain the strong asymptotics for the sequence of monic polynomials minimizing the norm

     

Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights

We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form , with γ>0, which include as particular cases the counterparts of the so-called Freud (i.e., when φ has a polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived.     

Some inequalities for symmetric functions and an application to orthogonal polynomials

We present some sharp inequalities for symmetric functions and give an application to orthogonal polynomials.     

On five-diagonal Toeplitz matrices and orthogonal polynomials on the unit circle

This paper deals with modifications of the Lebesgue moment functional by trigonometric polynomials of degree 2 and their associated orthogonal polynomials on the unit circle. We use techniques of five-diagonal matrix factorization and matrix polynomials to study the existence of such orthogonal polynomials.Dedicated to Prof. Luigi Gatteschi on his 70th birthdayThis research was partially supported by Diputación General de Aragón under grant P CB-12/91.     

Riemann–Hilbert analysis for Jacobi polynomials orthogonal on a single contour

Classical Jacobi polynomials , with α,β>-1, have a number of well-known properties, in particular the location of their zeros in the open interval (-1,1). This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters α_n,β_n depend on n in such a way that

with . We restrict our attention to the case where the limits A,B are not both positive and take values outside of the triangle bounded by the straight lines A=0, B=0 and A+B+2=0. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential.The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift–Zhou steepest descent method based on the Riemann–Hilbert reformulation of Jacobi polynomials.     

Algorithms for the integration and derivation of Chebyshev series

General formulas for the mth integral and derivative of a Chebyshev polynomial of the first or second kind are presented. The result is expressed as a finite series of the same kind of Chebyshev polynomials. These formulas permit to accelerate the determination of such integrals or derivatives. Besides, it is presented formulas for the mth integral and derivative of finite Chebyshev series and a numerical algorithm for the direct evaluation of the mth derivative of such a series.