Families Of Orthogonal Polynomials Research Articles

On Markov’s theorem on zeros of orthogonal polynomials revisited

This paper briefly surveys Markov’s theorem related to zeros of orthogonal polynomials. Monotonicity of zeros of some families of orthogonal polynomials are reviewed in detail.

Nonintersecting Brownian bridges on the unit circle with drift

Nonintersecting Brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers. We investigate the effect of adding a drift term μ to walkers on the circle conditioned to start and end at the same position. For each return time T<π2 we show there exists a critical drift μc(T) such that if |μ−2πm/T|<μc(T) for some integer m then the expected winding number for each walker is asymptotically m. In addition, we compute the asymptotic distribution of total winding numbers in the double-scaling regime in which the expected number of walkers with winding number not equal to m is finite. The method of proof is Riemann–Hilbert analysis of a certain family of discrete orthogonal polynomials with varying complex exponential weights. This is the first asymptotic analysis of such a class of polynomials. We determine asymptotic formulas for these polynomials as the degree of the polynomial grows large and demonstrate the emergence of a second band of zeros by a mechanism not previously seen for discrete orthogonal polynomials with real weights.

ORTHOPOLY: A library for accurate evaluation of series of classical orthogonal polynomials and their derivatives

We present the ORTHOPOLY software that permits to evaluate, efficiently and accurately, finite series of any classical family of orthogonal polynomials (Chebyshev, Legendre, ultraspherical or Gegenbauer, Jacobi, Hermite and Laguerre orthogonal polynomials) and their derivatives. The basic algorithm is the BCS-algorithm (Barrio–Clenshaw–Smith derivative algorithm), that permits to evaluate the kth derivative of a finite series of orthogonal polynomials at any point without obtaining before the previous derivatives. Due to the presence of rounding errors, specially in the case of high order derivatives, we introduce the compensated BCS-algorithm, based on Error-Free Transformation techniques, that permits to relegate the influence of the conditioning of the problem up to second order in the round-off unit of the computer. The BCS and compensated BCS algorithms may also give running-error bounds to provide information about the accuracy of the evaluation process. The ORTHOPOLY software includes C and Matlab versions of all the algorithms, and they are designed to be easily used in longer softwares to solve physical, mathematical, chemical or engineering problems (illustrated on the Schrödinger equation for the radial hydrogen atom). Program summaryProgram Title:ORTHOPOLYProgram Files doi:http://dx.doi.org/10.17632/n55bpy5bsr.1Licensing provisions: GPLv3Programming language: C and Matlab versionsNature of problem: Accurate numerical evaluation of finite series of classical orthogonal polynomials and their derivatives.Solution method:Barrio–Clenshaw–Smith algorithm for the evaluation of derivatives of finite series of classical orthogonal polynomials. Error-Free Transformation techniques for the Compensated Barrio–Clenshaw–Smith algorithm in order to provide accurate evaluations. Running-error techniques to provide error bounds of the evaluations.

Worst-Case Examples for Lasserre’s Measure–Based Hierarchy for Polynomial Optimization on the Hypercube

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre, and a related hierarchy by de Klerk, Hess, and Laurent. For polynomial optimization over the hypercube, we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials.

Approximate semi-analytical solution for a penny-shaped rough-walled hydraulic fracture driven by turbulent fluid in an impermeable rock

The popularity of high injection rate hydraulic fracturing treatments using low viscosity fluids is driving a need to consider the turbulent and laminar-turbulent transition regimes of fluid flow in hydraulic fracture simulators. The radial model is one of the most important geometries both for benchmarking and as a starter solution for 3D and Planar 3D models. Here we provide a semi-analytical, orthogonal polynomial series solution for a rough-walled radial (penny-shaped) hydraulic fracture driven by a fully turbulent fluid. Embedding the appropriate pressure singularities in a family of orthogonal polynomials used for derivation of the solution leads to very rapid convergence of the series, requiring just two terms for an accurate result. We conclude with an investigation of the occurrence of this limiting regime by comparison with numerical simulations, illustrating that the fully turbulent regime is typically not encountered for the radial geometry, although the present solution remains necessary as a starter solution and benchmark for the numerical simulators that are required to capture the laminar-turbulent transition. By comparison with numerical simulations that consider the laminar-turbulent transition, we find that such an estimate is expected to be sufficient for practical purposes when the inlet opening predicted by the turbulent solution exceeds the inlet opening predicted by the laminar solution.

Laguerre-Freud equations for Generalized Hahn polynomials of type I

We derive a system of difference equations satisfied by the three-term recurrence coefficients of some families of discrete orthogonal polynomials.

Hurwitz Polynomials and Orthogonal Polynomials Generated by Routh–Markov Parameters

A new criterion of Hurwitzness of odd-degree polynomials in terms of Markov parameters is given. Relationships between a family of orthogonal polynomials performed by Routh–Markov parameters and a family of Hurwitz polynomials of descending degree generated by the stability test are attained.

Root power sums and Chebyshev polynomials

We determine sequences of polynomials with rational coefficients that have certain postulated values for their root power sums. These, in turn, determine four families of orthogonal polynomials that can be expressed in terms of Chebyshev polynomials by a change of variables. We examine properties of these sequences of polynomials, two of which have already been studied in a previous paper.

A two-pronged progress in structured dense matrixvector multiplication.

Matrix-vector multiplication is one of the most fundamental computing primitives. Given a matrix and a vector , it is known that in the worst case Θ(N 2) operations over F are needed to compute Ab. Many types of structured matrices do admit faster multiplication. However, even given a matrix A that is known to have this property, it is hard in general to recover a representation of A exposing the actual fast multiplication algorithm. Additionally, it is not known in general whether the inverses of such structured matrices can be computed or multiplied quickly. A broad question is thus to identify classes of structured dense matrices that can be represented with O(N) parameters, and for which matrix-vector multiplication (and ideally other operations such as solvers) can be performed in a sub-quadratic number of operations. One such class of structured matrices that admit near-linear matrix-vector multiplication are the orthogonal polynomial transforms whose rows correspond to a family of orthogonal polynomials. Other well known classes include the Toeplitz, Hankel, Vandermonde, Cauchy matrices and their extensions (e.g. confluent Cauchy-like matrices) that are all special cases of a low displacementrank property. In this paper, we make progress on two fronts: Our work unifies, generalizes, and simplifies existing state-of-the-art results in structured matrix-vector multiplication. Finally, we show how applications in areas such as multipoint evaluations of multivariate polynomials can be reduced to problems involving low recurrence width matrices.

Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion

We study the approximation and stability properties of a recently popularized discretization strategy for the speed variable in kinetic equations, based on pseudo-spectral collocation on a grid defined by the zeros of a non-standard family of orthogonal polynomials called Maxwell polynomials. Taking a one-dimensional equation describing energy diffusion due to Fokker–Planck collisions with a Maxwell–Boltzmann background distribution as the test bench for the performance of the scheme, we find that Maxwell based discretizations outperform other commonly used schemes in most situations, often by orders of magnitude. This provides a strong motivation for their use in high-dimensional gyrokinetic simulations. However, we also show that Maxwell based schemes are subject to a non-modal time stepping instability in their most straightforward implementation, so that special care must be given to the discrete representation of the linear operators in order to benefit from the advantages provided by Maxwell polynomials.

Computation of induced orthogonal polynomial distributions

We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad class of measures, encompassing those associated to classical orthogonal polynomial families, which is stable for polynomial degrees up to at least 1000. Paired with other standard tools such as a numerical root-finding algorithm and inverse transform sampling, this provides a methodology for generating random samples from an induced orthogonal polynomial measure. Generating samples from this measure is one ingredient in optimal numerical methods for certain types of multivariate polynomial approximation. For example, sampling from induced distributions for weighted discrete least-squares approximation has recently been shown to yield convergence guarantees with a minimal number of samples. We also provide publicly-available code that implements the algorithms in this paper for sampling from induced distributions.

Cubic decomposition of a family of semiclassical orthogonal polynomials of class two

ABSTRACTWe deal with a family of semiclassical orthogonal polynomial sequences of class two having the cubic decomposition , . Only four monic orthogonal polynomial sequences (MOPS) appear in which their recurrence coefficients are explicitly given.

Macdonald–Koornwinder moments and the two-species exclusion process

Introduced in the late 1960’s (Macdonald et al. in Biopolymers 6:1–25, 1968; Spitzer in Adv Math 5:246–290, 1970), the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice with open boundaries. It has been known for awhile that there is a tight connection between the partition function of the ASEP and moments of Askey–Wilson polynomials (Uchiyama et al. in J Phys A 37(18):4985–5002, 2004; Corteel and Williams in Duke Math J 159(3):385–415, 2011; Corteel et al. in Trans Am Math Soc 364(11):6009–6037, 2012), a family of orthogonal polynomials which are at the top of the hierarchy of classical orthogonal polynomials in one variable. On the other hand, Askey–Wilson polynomials can be viewed as a specialization of the multivariate Macdonald–Koornwinder polynomials (also known as Koornwinder polynomials), which in turn give rise to the Macdonald polynomials associated to any classical root system via a limit or specialization (van Diejen in Compos Math 95(2):183–233, 1995). In light of the fact that Koornwinder polynomials generalize the Askey–Wilson polynomials, it is natural to ask whether one can find a particle model whose partition function is related to Koornwinder polynomials. In this article we answer this question affirmatively, by showing that Koornwinder moments at $$q=t$$ are closely connected to the partition function for the two-species exclusion process.

Generating functions and a family of special orthogonal polynomials

ABSTRACTIn this paper, we use quadratic decomposition to develop a method for constructing generating functions for orthogonal polynomials verifying the three-term recurrence relation with , . As application, we can deduce explicitly the expression for the generating function in some well-known semiclassical cases.

Differential equations for families of semi-classical orthogonal polynomials within class one

In this paper we study families of semi-classical orthogonal polynomials within class one. We derive general second or third order ordinary differential equations (with respect to certain parameters) for the recurrence coefficients of the three-term recurrence relation of these polynomials and show that in particular well-known cases, e.g. related to the modified Airy and Laguerre weights, these equations can be reduced to the second and the fourth Painlevé equations.

Divided-difference equation and three-term recurrence relations of some systems of bivariate q-orthogonal polynomials

Partial divided-difference equations and three-term recurrence relations satisfied by the bivariate Askey–Wilson and the bivariate q-Racah polynomials are computed in this work. By using limiting processes, partial divided (or q)-difference equations and three-term recurrence relations are also provided for each of the following families of orthogonal polynomials: the bivariate continuous dual q-Hahn, the bivariate Al-Salam-Chihara, the bivariate continuous q-Hahn, the bivariate q-Hahn, the bivariate dual q-Hahn, the bivariate q-Krawtchouk, the bivariate q-Meixner, and the bivariate q-Charlier polynomials.

Combinatorics of the two-species ASEP and Koornwinder moments

In previous work [12–14], the first and third authors introduced staircase tableaux, which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process (ASEP) and for the moments of the Askey–Wilson weight function. The fact that the ASEP and Askey–Wilson moments are related at all is unexpected, and is due to [45]. The ASEP is a model of particles hopping on a one-dimensional lattice of N sites with open boundaries; particles can enter and exit at both left and right borders. It was introduced around 1970 [34,43] and is cited as a model for both traffic flow and translation in protein synthesis. Meanwhile, the Askey–Wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials. So from this previous work, we have the relationship ASEP −− staircase tableaux −− Askey–Wilson moments. The Askey–Wilson polynomials can be viewed as the one-variable case of the multivariate Koornwinder polynomials, which are also known as the Macdonald polynomials attached to the non-reduced affine root system (Cn∨,Cn). It is natural then to ask whether one can generalize the relationships among the ASEP, Askey–Wilson moments, and staircase tableaux, in such a way that Koornwinder moments replace Askey–Wilson moments. In [15], we made a precise link between Koornwinder moments and the two-species ASEP, a generalization of the ASEP which has two species of particles with different “weights.” In this article we introduce rhombic staircase tableaux, and show that we have the relationship 2-species ASEP −− rhombic staircase tableaux −− Koornwinder moments. In particular, we give formulas for the stationary distribution of the two-species ASEP and for Koornwinder moments, in terms of rhombic staircase tableaux. [Display omitted]

Application of modal filtering to a spectral difference method

We adapt the spectral viscosity (SV) formulation implemented as a modal filter to a Spectral Difference Method (SD) solving hyperbolic conservation laws. In the SD Method we use selections of different orthogonal polynomials (APK polynomials). Furthermore we obtain new error bounds for filtered APK extensions of smooth functions. We demonstrate that the modal filter also depends on the chosen polynomial basis in the SD Method. Spectral filtering stabilizes the scheme and leaves weaker oscillations. Hence, the selection of the family of orthogonal polynomials on triangles and their specific modal filter possesses a positive influence on the stability and accuracy of the SD Method. In the second part, we initiate a stability analysis for a linear scalar test case with periodic initial condition to find the best selection of APK polynomials and their specific modal filter. To the best of our knowledge, this work is the first that gives a stability analysis for a scheme with spectral filtering. Finally, we demonstrate the influence of the underlying basis of APK polynomials in a well-known test case.

A family of discrete semi-classical orthogonal polynomials of class one

In this work, we solve the system of Laguerre–Freud equations for the recurrence coefficients $$\beta _n$$ , $$\gamma _{n+1} , n \ge 0$$ of the $$D_{w}$$ -semi-classical orthogonal polynomials sequences of class one in the case when $$\beta _{0}=-t_{0}$$ , $$\beta _{n+1}=t_{n}-t_{n+1}$$ and $$\gamma _{n+1}=-t_{n}^{2}$$ with $$t_{n}\ne 0\;n\ge 0$$ , where $$D_w$$ is the divided difference operator. There are essentially four canonical families.

New discrete orthogonal moments for signal analysis

The paper addresses some computational aspects and applications of the polynomial unbiased finite impulse response functions originally derived by Shmaliy as a new class of a one-parameter family of discrete orthogonal polynomials. We present two new explicit formulas to compute these polynomials directly. They are based on the discrete Rodrigues’ representation and hypergeometric form, respectively. These straightforward calculations lead us to propose another discrete signal representation in moment domain. Experimental results show a comparison between the proposed moments and discrete Chebyshev moments in terms of noise-free/noisy signal feature extraction and reconstruction error.