Continuous Orthogonal Polynomials Research Articles

Basic theory of finite-type relations between orthogonal polynomial sequences

A theory of finite-type relations between two orthogonal polynomial sequences is developed. In fact, we find necessary and sufficient conditions for the extended connection coefficients which should be fulfilled for a finite-type relation. Next, a simplification criterion is given and some applications in the domain of continuous, discrete and quantum semiclassical orthogonal polynomials are treated into detail.

Orthogonal Polynomials in Mathematical Physics

This is a review of ([Formula: see text]-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials, and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of ([Formula: see text]-)hypergeometric functions and their integral representations. In particular, this gives rise to relations with conformal blocks of the Virasoro algebra. To the memory of Ludwig Dmitrievich Faddeev

Random surface growth and Karlin-McGregor polynomials

We consider consistent dynamics for non-intersecting birth and death chains, originating from dualities of stochastic coalescing flows and one dimensional orthogonal polynomials. As corollaries, we obtain unified and simple probabilistic proofs of certain key intertwining relations between multivariate Markov chains on the levels of some branching graphs. Special cases include the dynamics on the Gelfand-Tsetlin graph considered in the seminal work of Borodin and Olshanski in [10] and the ones on the BC-type graph recently studied by Cuenca in [17]. Moreover, we introduce a general inhomogeneous random growth process with a wall that includes as special cases the ones considered by Borodin and Kuan [8] and Cerenzia [15], that are related to the representation theory of classical groups and also the Jacobi growth process more recently studied by Cerenzia and Kuan [16]. Its most important feature is that, this process retains the determinantal structure of the ones studied previously and for the fully packed initial condition we are able to calculate its correlation kernel explicitly in terms of a contour integral involving orthogonal polynomials. At a certain scaling limit, at a finite distance from the wall, one obtains for a single level discrete determinantal ensembles associated to continuous orthogonal polynomials, that were recently introduced by Borodin and Olshanski in [11], and that depend on the inhomogeneities.

Generalizations of generating functions for higher continuous hypergeometric orthogonal polynomials in the Askey scheme

We use connection relations and series rearrangement to generalize generating functions for several higher continuous orthogonal polynomials in the Askey scheme, namely the Wilson, continuous dual Hahn, continuous Hahn, and Meixner–Pollaczek polynomials. We also determine corresponding definite integrals using the orthogonality relations for these polynomials.

The Painlevé Equations and Orthogonal Polynomials

Abstract This article is a survey of the recent studies jointly with Lies Boelen, Christophe Smet, Walter Van Assche and Lun Zhang (KULeuven, Belgium) on semi-classical continuous and discrete orthogonal polynomials and, in particular, on the connection of their recurrence coefficients to the solutions of the Painleve equations. After recalling some basic facts about the Painleve equations, we discuss continuous and discrete orthogonal polynomials and explain their connection.

Generalised Christoffel–Darboux formula most directly applied in generating fully symmetric doubly resistively terminated LC lossless ladder filters

In this article, the approximation problem of a continuous analogue filter function of even and odd order is solved mathematically most directly applying the proposed generalised Christoffel–Darboux formula for two continuous orthogonal polynomials (Chebyshev first and second kind) on the equal finite segment of orthogonality in a compact explicit representative form. A set of analytic expressions of the proposed formula for the representative examples of odd-orders is given. Additionally, these expressions are applied in generating new excellent class all pole low-pass prototype analogue filter functions of odd orders (7, 9, 11, and 13th order). Based on the generated functions, the fully symmetric doubly resistively terminated LC ladder filter networks are realised by Darlington realisation. The generated filters are analysed, and their characteristics are presented. The effects of finite tolerance of a critical reactance of these ladder filters on the filter characteristics are considered. The filters generated by the proposed formula show excellent properties and possess important advantages in comparison to the corresponding filters generated by other methods.

Fast computation of orthogonal Fourier–Mellin moments in polar coordinates

Fast, accurate and memory-efficient method is proposed for computing orthogonal Fourier---Mellin moments. Since the basis polynomials are continuous orthogonal polynomials defined in polar coordinates over a unit disk, the proposed method is applied to polar coordinates where the unit disk is divided into a number of non-overlapping circular rings that are divided into circular sectors of the same area. Each sector is represented by one point in its center. The implementation of this method completely removes both approximation and geometrical errors produced by the conventional methods. Based on the symmetry property, a fast and memory-efficient algorithm is proposed to accelerate the moment's computations. A comparison to conventional methods is performed. Numerical experiments are performed to ensure the efficiency of the proposed method.

Dual‐based error analysis for uncertainty quantification in a chemical system

AbstractAn a posteriori error estimation for the numerical solution of a stochastic variational problem arising in the context of parametric uncertainty is presented. The discretization of the stochastic variational problem uses piecewise continuous orthogonal polynomials in the stochastic domain. The a posteriori methodology is derived by measuring the error as the functional difference between the continuous and discrete solutions. An example is shown on a chemical system which reaction rates are uncertain. The present approach efficiency in providing a good accuracy in the uncertainty quantification at a reasonable computational cost is demonstrated. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Fast computation of accurate Zernike moments

Zernike polynomials are continuous orthogonal polynomials defined in polar coordinates over a unit disk. Zernike moment’s computation using conventional methods produced two types of errors namely approximation and geometrical. Approximation errors are removed by using exact Zernike moments. Geometrical errors are minimized through a proper mapping of the image. Exact Zernike moments are expressed as a combination of exact radial moments, where exact values of radial moments are computed by mathematical integration of the monomial polynomials over digital image pixels. A fast algorithm is proposed to accelerate the moment’s computations. A comparison with other conventional methods is performed. The obtained results explain the superiority of the proposed method.

The Reproducing Kernel Structure Arising from a Combination of Continuous and Discrete Orthogonal Polynomials into Fourier Systems

We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier-type systems.We prove Ismail’s conjecture regarding the existence of a reproducing kernel structure behind these kernels, by establishing a link with Saitoh’s theory of linear transformations in Hilbert space. The results are illustrated with Fourier kernels with ultraspherical, their continuous q-extensions and generalizations. As a byproduct of this approach, a new class of sampling theorems is obtained, as well as Neumann-type expansions in Bessel and q-Bessel functions.

Diophantine equations for classical continuous orthogonal polynomials

Let A, B, C denote rational numbers with AB ≠ 0 and m > n ≥ 3 arbitrary rational integers. We study the Diophantine equation AP m ( x) + Bp n ( y) = C, in x, y ϵ Z , where { Pk( x)}I is one of the three classical continuous orthogonal polynomial families, i.e. Laguerre polynomials, Jacobi polynomials (including Gegenbauer, Legendre or Chebyshev polynomials) and Hermite polynomials. We prove that with exception of the Chebyshev polynomials for all such polynomial families there are at most finitely many solutions ( x, y) ϵ Z 2 provided n > 4. The tools are besides the criterion [3], a theorem of Szeg— [14] on monotonicity of stationary points of polynomials which satisfy a second order Sturm-Liouville differential equation, (ax 2 + bx + c)y″ n(x) + (dx + e)y′(x) - λ ny n(x) = 0, n ϵ Z≥0.

Fourth-order difference equation satisfied by the co-recursive of q-classical orthogonal polynomials

We derive the fourth-order q-difference equation satisfied by the co-recursive of q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials φ and ψ appearing in the q-Pearson difference equation D q(φρ)=ψρ defining the weight ρ of the q-classical orthogonal polynomials inside the q-Hahn tableau. Use of suitable change of variable and limit processes allow us to recover the results known for the co-recursive of the classical continuous and classical discrete orthogonal polynomials. Moreover, we describe particular situations for which the co-recursive of classical orthogonal polynomials are still classical and express these new families in terms of the starting ones.

Entropic integrals of orthogonal hypergeometric polynomials with general supports

The Boltzmann–Shannon information entropy of probability measures which involve the continuous hypergeometric-type polynomials { p n ( x)}, orthogonal with respect to a general weight function ω( x), is determined by two integral quantities: one with kernel p n 2(x)ω(x) ln p n 2(x) , called as entropy of the polynomial p n ( x), and another one with kernel p n 2(x)ω(x) ln ω(x) . Here, an explicit expression for the latter quantity, and for a broader family of related integrals, is obtained in terms only of the second-order differential equation satisfied by the involved polynomials. For illustration, the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type (Hermite, Laguerre, and Jacobi).

The Fourth-order Difference Equation Satisfied by the Associated Orthogonal Polynomials of the Δ -Laguerre–Hahn Class

Starting from the Dω-Riccati difference equation satisfied by the Stieltjes function of a linear functional, we work out an algorithm which enables us to write the general fourth-order difference equation satisfied by the associated of any integer order of orthogonal polynomials of the Δ -Laguerre–Hahn class. Moreover, in classical situations (Meixner, Charlier, Krawtchouk and Hahn), we give these difference equations explicitly; and from the Hahn difference equation, by limit processes we recover the difference equations satisfied by the associated of the classical discrete orthogonal polynomials and the differential equations satisfied by the associated of the classical continuous orthogonal polynomials.

Transverse limits in the Askey tableau

Limit relations between classical discrete (Charlier, Meixner, Kravchuk, Hahn) to classical continuous (Jacobi, Laguerre, Hermite) orthogonal polynomials (to be called transverse limits) are nicely presented in the Askey tableau and can be described by relations of type lim ω →0 P n ( x( s, ω)) = Q n ( s), where P n ( s) and Q n ( s) stand for the discrete and continuous family, respectively. Deeper information on these limiting processes can be obtained by expanding the discrete polynomial family as P n ( x9 s, ω)) = ∑ k=0 ∞ ω k R k ( s; n). In this paper a method for computing the coefficients R k ( s; n) is designed the main tool being the consideration of some closely related connection and inversion problems.

On the limit relations between classical continuous and discrete orthogonal polynomials

Limit relations between classical continuous (Jacobi, Laguerre, Hermite) and discrete (Charlier, Meixner, Kravchuk, Hahn) orthogonal polynomials are well known and can be described by relations of type lim λ → ∞ P n ( x; λ) = Q n ( x). Deeper information on these limiting processes can be obtained from the expansion P n(x;λ) = ∑ k=0 ∞ R k(x;n) λ k . In this paper a method for the recursive computation of coefficients R k ( x; n) is designed being the main tool the consideration of a closely related connection problem which can be solved, also recurrently, by using an algorithm recently developed by the authors.

Representations of orthogonal polynomials

Zeilberger's algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computers recurrence and differential equations for hyperexponential integrals. Further versions of this algorithm allow the computation of recurrence and differential equations from Rodrigues type formulas and from generating functions. In particular, these algorithms can be used to compute the differential/difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations, and from their Rodrigues representations and generating functions. In recent work, we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials, in terms of the coefficients of their differential/difference equations, to give an algorithm to identify the polynomial system from a given recurrence equation. In this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential/difference equation, the hypergeometric representation, and the recurrence equation is possible. The main technique is again to use explicit formulas for structural identities of the given polynomial systems.

Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case

We present a simple approach in order to compute recursively the connection coefficients between two families of classical (continuous) orthogonal polynomials (Hermite, Laguerre, Jacobi and Bessel), i.e. the coefficients C m ( n) in the expression P n(x) = ∑ m=0 n (m (n)Q m(x) , where { P n ( x)} and { Q m ( x)} belong to the aforementioned class of polynomials. This is done by adapting a general and systematic algorithm, recently developed by the authors, to the classical situation. Moreover, extensions of this method to some related connection problems and to the generalized linearization problem are given.

Results for some inversion problems for classical continuous and discrete orthogonal polynomials

Explicit expressions for the coefficients in the expansion of classical discrete orthogonal polynomials (Charlier, Meixner, Krawtchouck, Hahn, Hahn - Eberlein) into the falling factorial basis are given. The corresponding inversion problems are solved explicitly. This is done by using a general algorithm, recently developed by the authors, which is also applied to this kind of inversion problem but relating the basis and the classical (continuous) orthogonal polynomials of Jacobi, Laguerre, Hermite and Bessel.

On a general q-Fourier transformation with nonsymmetric kernels

Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a q-Fourier transform by using the continuous q-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey-Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of q-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey-Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.