Characterization Of Orthogonal Polynomials Research Articles

Characterization of orthogonal polynomials on lattices

We consider two sequences of orthogonal polynomials and such that with , and are sequences of complex numbers, , is the identity operator, x defines a lattice, and . We show that under some natural conditions, both involved orthogonal polynomials sequences and are semiclassical whenever k = m. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices.

On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices

In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or q-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the “left” inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution—non polynomial solution—of a second-order divided-difference equation of hypergeometric type.

Connection relations and characterizations of orthogonal polynomials

We give a general method of characterizing symmetric orthogonal polynomials through a certain type of connection relations. This method is applied to Al-Salam–Chihara, Askey–Wilson, and Meixner–Pollaczek polynomials. This characterization technique unifies and extends some previous characterization results of Lasser and Obermaier and Ismail and Obermaier. Along the way we explicitly evaluate the connection coefficients in the expansion of Dq2pn in terms of {pk}, where Dq is the Askey–Wilson operator and {pk} are general Askey–Wilson polynomials. As a limiting case we derive the corresponding connection coefficients in the expansion of W2Wn in terms of {Wk}, where W is the Wilson operator and {Wk} are general Wilson polynomials. Using the connection relation for Askey–Wilson polynomials, we obtain a characterization for the two-parameter symmetric Askey–Wilson polynomials. The connection relations between DmPn(α,β), D:=d/dx and {Pk(α,β)} are also derived.

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c ( x i ) = c i for i = 0 , 1 , … . We proved that the nonlinear functional equations P ( t ) = c ( P ( x ) P ( α x + t ) ) and P ( t ) = c ( P ( x ) P ( x t ) ) 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.

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) = xα 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.

Universal Behavior for Averages of Characteristic Polynomials at the Origin of the Spectrum

It has been shown by Strahov and Fyodorov that averages of products and ratios of characteristic polynomials corresponding to Hermitian matrices of a unitary ensemble, involve kernels related to orthogonal polynomials and their Cauchy transforms. We will show that, for the unitary ensemble Hermitian matrices, these kernels have universal behavior at the origin of the spectrum, as n→∞, in terms of Bessel functions. Our approach is based on the characterization of orthogonal polynomials together with their Cauchy transforms via a matrix Riemann-Hilbert problem, due to Fokas, Its and Kitaev, and on an application of the Deift/Zhou steepest descent method for matrix Riemann-Hilbert problems to obtain the asymptotic behavior of the Riemann-Hilbert problem.

Characterization of orthogonal polynomials with respect to a functional

In this paper we study questions of existence, uniqueness and characterization of polynomials orthogonal with respect to a linear, not necessarily definite, functional L defined on the set of Laurent polynomials. First we characterize with the help of the function F L(z):= L( (y + z) (y − z) ) polynomials orthogonal with respect to L . Using this characterization, which has wide applications, we are able to settle the question of existence and uniqueness of orthogonal polynomials. The uniquely determined orthogonal polynomials will be called basic orthogonal polynomials. It is to be pointed out that, in contrast to the real case, there are natural numbers n μ such that there exist no polynomials of degree n μ which are orthogonal with respect to L , if L is indefinite and if we have “orthogonality-jumps” greater than 1. Furthermore the functional to which the basic orthogonal polynomials of the second kind are orthogonal is determined. Finally, we get explicit expressions for all basic orthogonal polynomials with respect to a “weight function” the support of which consists of several arcs of the unit circle, changes sign from arc to arc and has square root singularities at the boundary points of the arcs. These polynomials can be considered as the basic polynomials in describing and generating orthogonal polynomials with periodic reflection coefficients.

Orthogonal matrix polynomials and higher-order recurrence relations

It is well known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel measure on the real line. We extend this result and show that every system of polynomials satisfying some (2 N+1)-term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are N × N matrices. We apply this result to polynomials orthogonal with respect to a discrete Sobolev inner product and other inner products in the linear space of polynomials. As an application we give a short proof of Krein's characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points.

Characterizations of Orthogonal Polynomials Satisfying Differential Equations

In 1938, H. L. Krall found a necessary and sufficient condition for an orthogonal polynomial set $\{ P_n (x)\} _0^\infty $ to satisfy a linear differential equation of the form \[ \sum_{i = 0}^N {\ell _i (x)y^{(i)} } (x) = \lambda _n y(x).\] Here the authors give a new simple proof of Krall’s theorem as well as some other characterizations of such orthogonal polynomial sets based on the symmetrizability of the differential operator. In particular it is shown that such orthogonal polynomial sets are characterized by a certain Sobolev-type orthogonality, which generalizes Hahn’s charaterization of classical orthogonal polynomials.

A characterization of orthogonal polynomials

A characterization of orthogonal polynomials