Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

On Sobolev Orthogonality for the Generalized Laguerre Polynomials

The orthogonality of the generalized Laguerre polynomials, {L^(α)_n(x)}_n0, is a well known fact when the parameterαis a real number but not a negative integer. In fact, for −1<α, they are orthogonal on the interval [0, +∞) with respect to the weight functionρ(x)=x^αe^−x, and forα<−1, but not an integer, they are orthogonal with respect to a non-positive definite linear functional. In this work we will show that, for every value of the real parameterα, the generalized Laguerre polynomials are orthogonal with respect to a non-diagonal Sobolev inner product, that is, an inner product involving derivatives.     

The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]

We consider polynomials that are orthogonal on [−1,1] with respect to a modified Jacobi weight (1−x)^α(1+x)^βh(x), with α,β>−1 and h real analytic and strictly positive on [−1,1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [−1,1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [−1,1]. For the asymptotic analysis we use the steepest descent technique for Riemann-Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szeg? function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions.     

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w_n(x)dx = e^−nV(x) dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [19, 20]. The Riemann‐Hilbert problem is analyzed in turn using the steepest‐descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμ_V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.     

A Two-Parameter Family of Orthogonal Polynomials with Respect to a Jacobi-Type Weight on the Unit Circle

A two-parameter family of polynomials is introduced by a recursion formula. The polynomials are orthogonal on the unit circle with respect to the weight ω_{α, β}(θ) = |(1 − z)^α(1 + z)^β|², α, β > − , z = e^iθ. Explicit representation, norm estimates, shift identities, and explicit connection to Jacobi polynomials on the real interval [−1, 1] is presented.     

A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

In this paper we deal with a family of nonstandard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so-called Δ-Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the Δ-Meixner–Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre–Sobolev orthogonal polynomials.     

Asymptotics for polynomials orthogonal over the unit disk with respect to a positive polynomial weight

We derive asymptotics for polynomials orthogonal over the complex unit disk with respect to a weight of the form ²|h(z)|, with h(z) a polynomial without zeros in |z|<1. The behavior of the polynomials is established at every point of the complex plane. The proofs are based on adapting to the unit disk a technique of J. Szabados for the asymptotic analysis of polynomials orthogonal over the unit circle with respect to the same type of weight.     

Strong asymptotics of orthogonal polynomials with respect to exponential weights

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e^−Q(x) dx on the real line, where Q(x) = Σ q_k x^k, q_2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [22, 23]. We employ the steepest‐descent‐type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel‐Rotach‐type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. © 1999 John Wiley & Sons, Inc.     

Orthogonal Structure on a Wedge and on the Boundary of a Square

Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier orthogonal expansions is studied. These are used to establish analogous results for orthogonal polynomials on the boundary of the square. As an application, we study the statistics of the associated determinantal point process and use the basis to calculate Stieltjes transforms.

     

Construction of polynomials that are orthogonal with respect to a discrete bilinear form

We describe a new algorithm for the computation of recursion coefficients of monic polynomials {p _j } _j ^=0/n that are orthogonal with respect to a discrete bilinear form (f, g) := _k ^=1/m f(x _k )g(x _k )w _k ,m n, with real distinct nodesx _k and real nonvanishing weightsw _k . The algorithm proceeds by applying a judiciously chosen sequence of real or complex Givens rotations to the diagonal matrix diag[x ₁,x ₂, ...,x _m ] in order to determine an orthogonally similar complex symmetric tridiagonal matrixT, from whose entries the recursion coefficients of the monic orthogonal polynomials can easily be computed. Fourier coefficients of given functions can conveniently be computed simultaneously with the recursion coefficients. Our scheme generalizes methods by Elhay et al. [6] based on Givens rotations for updating and downdating polynomials that are orthogonal with respect to a discrete inner product. Our scheme also extends an algorithm for the solution of an inverse eigenvalue problem for real symmetric tridiagonal matrices proposed by Rutishauser [20], Gragg and Harrod [17], and a method for generating orthogonal polynomials based theoron [18]. Computed examples that compare our algorithm with the Stieltjes procedure show the former to generally yield higher accuracy except whenn m. Ifn is sufficiently much smaller thanm, then both the Stieltjes procedure and our algorithm yield accurate results.Research supported in part by the Center for Research on Parallel Computation at Rice University and NSF Grant No. DMS-9002884.     

On sieved orthogonal polynomials. VIII. Sieved associated Pollaczek polynomials

We study a general orthogonal polynomial set which includes the sieved associated ultraspherical and the sieved Pollaczek polynomials. This we get by letting q approach a root of unity in the recurrence relation and the generating functions of the associated q-ultraspherical and the Pollaczek polynomials. We find the weight functions with respect to which these polynomials are orthogonal and determine the asymptotic behavior of these polynomials on and off their interval of orthogonality.     

Relativistic Lamé functions: Completeness vs. polynomial asymptotics

In earlier work we introduced and studied two commuting generalized Lamé operators, obtaining in particular joint eigenfunctions for a dense set in the natural parameter space. Here we consider these difference operators and their eigenfunctions in relation to the Hilbert space L²((0, π/r), w(x)dx), with r > 0 and the weight function w(x) a ratio of elliptic gamma functions. In particular, we show that the previously known pairwise orthogonal joint eigenfunctions need only be supplemented by finitely many new ones to obtain an orthogonal base. This completeness property is derived by exploiting recent results on the large-degree Hilbert space asymptotics of a class of orthonormal polynomials. The polynomials p_n(cos(rx)), n ε , that are relevant in the Lamé setting are orthonormal in L²((0, π/r), w_P(x)dx), with w_p(x) closely related to w(x).     

Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e−φ(x), giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.     

Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = x^α e^-Q(x), where α > -1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach-type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coefficients and of the leading coefficients of the orthonormal polynomials. As an application we will use these asymptotics to prove universality results in random matrix theory. We will prove our results by using the characterization of orthogonal polynomials via a 2 × 2 matrix valued Riemann--Hilbert problem, due to Fokas, Its, and Kitaev, together with an application of the Deift-Zhou steepest descent method to analyze the Riemann-Hilbert problem asymptotically.     

Connection coefficients, orthogonal polynomials and the WZ-algorithms

In this paper we explore the relationship between the coefficients in the expansion of a function f(x) in orthogonal polynomials and the coefficients for the expansion of (1-x) ^m f(x), with particular attention to the case of Jacobi polynomials. Such problems arise frequently in computational chemistry. The analysis of the situation is substantially assisted by the use of two of the so-called Wilf-Zeilberger algorithms: the algorithm zeil and the algorithm hyper. We explain these algorithms and give several examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Gauss-Type Quadrature Rules

Some Gauss-type Quadrature rules over [0, 1], which involve values and/or the derivative of the integrand at 0 and/or 1, are investigated. Our work is based on the orthogonal polynomials with respect to linear weight function ω(t): = 1 ? t over [0, 1]. These polynomials are also linked with a class of recently developed “identity-type functions”. Along the lines of Golub's work, the nodes and weights of the quadrature rules are computed from Jacobi-type matrices with simple rational entries. Computational procedures for the derived rules are tested on different integrands. The proposed methods have some advantage over the respective Gauss-type rules with respect to the Gauss weight function ω(t): = 1 over [0, 1].     

An extension of a result of Zaharescu on irreducible polynomials

It is well known that if f(x) is a monic irreducible polynomial of degree d with coefficients in a complete valued field (K, ‖), then any monic polynomial of degree d over K which is sufficiently close to f(x) with respect to ‖ is also irreducible over K. In 2004, Zaharescu proved a similar result applicable to separable, irreducible polynomials over valued fields which are not necessarily complete. In this paper, the authors extend Zaharescu’s result to all irreducible polynomials without assuming separability.     

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.     

Quasi-<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm orthogonal Galerkin expansions in sums of Jacobi polynomials

In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 − x) ^α (1 + x) ^β . However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an L ^p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.     

A method of constructing Krall's polynomials

We propose a method of constructing orthogonal polynomials P_n(x) (Krall's polynomials) that are eigenfunctions of higher-order differential operators. Using this method we show that recurrence coefficients of Krall's polynomials P_n(x) are rational functions of n. Let P_n^(a,b;M)(x) be polynomials obtained from the Jacobi polynomials P_n^(a,b)(x) by the following procedure. We add an arbitrary concentrated mass M at the endpoint of the orthogonality interval with respect to the weight function of the ordinary Jacobi polynomials. We find necessary conditions for the parameters a,b in order for the polynomials P_n^(a,b;M)(x) to obey a higher-order differential equation. The main result of the paper is the following. Let a be a positive integer and b⩾−1/2 an arbitrary real parameter. Then the polynomials P_n^(a,b;M)(x) are Krall's polynomials satisfying a differential equation of order 2a+4.