Recurrence Coefficients Of Orthogonal Polynomials Research Articles

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \bigl(\frac{2}{1-x}\bigr) {\rm d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by Magnus and extends earlier results by Conway and one of the authors. The proof relies on the Riemann-Hilbert method. The main difficulty in applying the method to the problem at hand is the lack of an appropriate local parametrix near the logarithmic singularity at $x = +1$.

Performance and accuracy of the basic closure algorithm of quadrature-based moment methods

Quadrature-based moment methods (QBMMs) are numerical methods to close moment equations derived from a population balance equation. There are many variants, most of which are based on the same closure algorithm that can be divided into three subroutines: First, computing the recurrence coefficients of orthogonal polynomials from a moment sequence, second, finding the associated Gaussian quadrature rule by solving an eigenvalue problem, and third, closing the moment equations using that quadrature rule. Since each of these steps may be carried out billions of times in large-scale simulations, knowing the potentials of optimization is the key to making QBMMs as efficient as possible. In this work, we investigate in detail the performance and accuracy of QBMMs with different configurations and over a wide range of input moment sets. Our major findings were the following: Available algorithms to compute the recurrence coefficients are about equally sensitive to the input moment sequence. There are considerable differences in performance, but the contribution to the overall computational costs of the closure algorithm is negligible. The standard method to solve the eigenvalue problem is insensitive to the input moment set in contrast to a less robust but slightly faster alternative that involves solving a Vandermonde system. For relatively simple physical problems, the overall computational costs are mostly determined by the eigenvalue problem, whereas closing the moment equations becomes important in the presence of more complex processes with particle-particle interactions. The complete source code we implemented and used for our numerical investigations is publicly available.

The Recurrence Coefficients of Orthogonal Polynomials with a Weight Interpolating between the Laguerre Weight and the Exponential Cubic Weight

In this paper, we consider the orthogonal polynomials with respect to the weight w(x)=w(x;s):=xλe−N[x+s(x3−x)],x∈R+, where λ>0, N>0 and 0≤s≤1. By using the ladder operator approach, we obtain a pair of second-order nonlinear difference equations and a pair of differential–difference equations satisfied by the recurrence coefficients αn(s) and βn(s). We also establish the relation between the associated Hankel determinant and the recurrence coefficients. From Dyson’s Coulomb fluid approach, we prove that the recurrence coefficients converge and the limits are derived explicitly when q:=n/N is fixed as n→∞.

Differential and difference equations for recurrence coefficients of orthogonal polynomials with a singularly perturbed Laguerre-type weight

We are concerned with the monic orthogonal polynomials with respect to a singularly perturbed Laguerre-type weight. By using the ladder operator approach, we derive a complicated system of nonlinear second-order difference equations satisfied by the recurrence coefficients. This allows us to derive the large n n asymptotic expansions of the recurrence coefficients. In addition, we also obtain a system of differential-difference equations for the recurrence coefficients.

A generalized sextic Freud weight

ABSTRACT We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalized sextic Freud weight with parameters and . We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalized hypergeometric functions . We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalized quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalized hypergeometric functions.

Unique positive solutions to q-discrete equations associated with orthogonal polynomials

Boelen et al. [q-discrete Painlevé equations for recurrence coefficients of modified q-Freud orthogonal polynomials, J. Differ. Equ. Appl. 16(1) (2010), pp. 37–53] deduced a q-discrete Painlevé equation satisfied by the recurrence coefficients of orthogonal polynomials and conjectured that the equation had a unique positive solution. We prove their conjecture and discuss related work in the area, highlighting a potentially unifying methodology.

Gaussian unitary ensembles with two jump discontinuities, PDEs, and the coupled Painlevé II and IV systems

AbstractWe consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of Min and Chen [Math. Methods Appl Sci. 2019;42:301‐321] where a second‐order partial differential equation (PDE) was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system which was established in Wu and Xu [arXiv: 2002.11240v2] by a study of the Riemann‐Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as , the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé II system and it satisfies a second‐order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlevé II system.

Differential and difference equations for recurrence coefficients of orthogonal polynomials with hypergeometric weights and Bäcklund transformations of the sixth Painlevé equation

It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinear difference equations. There is also a connection to the solutions of the [Formula: see text]-form of the sixth Painlevé equation (one of the parameters of the weights being the independent variable in the differential equation) [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.]. In this paper, we derive a second-order nonlinear difference equation from the system and present explicit formulas showing how this difference equation arises from the Bäcklund transformations of the sixth Painlevé equation. We also present an alternative way to derive the connection between the recurrence coefficients and the solutions of the sixth Painlevé equation.

Polynomials orthogonal with respect to cardinal B-spline weight functions

A stable and efficient discretization procedure is developed to compute the recurrence coefficients for orthogonal polynomials whose weight function is a polynomial cardinal spline of order m ≥ 1. The procedure is compared with a symbolic moment-based method developed recently by G. V. Milovanovic. Numerical examples are provided for illustration.

On perturbed orthogonal polynomials on the real line and the unit circle via Szegő’s transformation

By using the Szegő’s transformation we deduce new relations between the recurrence coefficients for orthogonal polynomials on the real line and the Verblunsky parameters of orthogonal polynomials on the unit circle. Moreover, we study the relation between the corresponding S-functions and C-functions.

A Generalized Freud Weight

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight urn:x-wiley:00222526:media:sapm12105:sapm12105-math-0001with parameters and , and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlevé equation. Further we derive a second‐order linear ordinary differential equation and a differential‐difference equation satisfied by the generalized Freud polynomials.

Computing recurrence coefficients of multiple orthogonal polynomials

Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a $(r+2)$-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence relation) and there is a system of $r$ recurrence relations connecting the nearest neighbors (the so-called nearest neighbor recurrence relations). In this paper we deal with two problems. First we show how one can obtain the nearest neighbor recurrence coefficients (and in particular the recurrence coefficients of the orthogonal polynomials for each of the defining measures) from the step-line recurrence coefficients. Secondly we show how one can compute the step-line recurrence coefficients from the recurrence coefficients of the orthogonal polynomials of each of the measures defining the multiple orthogonality.

The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlev\'e equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fourth Painlev\'e equation.

Caustics, counting maps and semi-classical asymptotics

This paper develops a deeper understanding of the structure and combinatorial significance of the partition function for Hermitian random matrices. The coefficients of the large N expansion of the logarithm of this partition function, also known as the genus expansion (and its derivatives), are generating functions for a variety of graphical enumeration problems. The main results are to prove that these generating functions are, in fact, specific rational functions of a distinguished irrational (algebraic) function, z0(t). This distinguished function is itself the generating function for the Catalan numbers (or generalized Catalan numbers, depending on the choice of weight of the parameter t). It is also a solution of the inviscid Burgers equation for certain initial data. The shock formation, or caustic, of the Burgers characteristic solution is directly related to the poles of the rational forms of the generating functions.As an intriguing application, one gains new insights into the relation between certain derivatives of the genus expansion, in a double-scaling limit, and the asymptotic expansion of the first Painlevé transcendent. This provides a precise expression of the Painlevé asymptotic coefficients directly in terms of the coefficients of the partial fractions expansion of the rational form of the generating functions established in this paper. Moreover, these insights point towards a more general program relating the first Painlevé hierarchy to the higher order structure of the double-scaling limit through the specific rational structure of generating functions in the genus expansion. The paper closes with a discussion of the relation of this work to recent developments in understanding the asymptotics of graphical enumeration.As a by-product, these results also yield new information about the asymptotics of recurrence coefficients for orthogonal polynomials with respect to exponential weights, the calculation of correlation functions for certain tied random walks on a 1D lattice, and the large time asymptotics of random matrix partition functions.

On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials

In 1995, Magnus [15] posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [ − 1 , 1 ] of the form ( 1 − x ) α ( 1 + x ) β | x 0 − x | γ × { B , for x ∈ [ − 1 , x 0 ) , A , for x ∈ [ x 0 , 1 ] , with A , B > 0 , α , β , γ > − 1 , and x 0 ∈ ( − 1 , 1 ) . We show rigorously that Magnus’ conjecture is correct even in a more general situation, when the weight above has an extra factor, which is analytic in a neighborhood of [ − 1 , 1 ] and positive on the interval. The proof is based on the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann–Hilbert problem characterizing the orthogonal polynomials. A feature of this situation is that the local analysis at x 0 has to be carried out in terms of confluent hypergeometric functions.

Variable-precision recurrence coefficients for nonstandard orthogonal polynomials

A symbolic/variable-precision procedure is described (and implemented in Matlab) that generates an arbitrary number N of recurrence coefficients for orthogonal polynomials to any given precision nofdig. The only requirement is the availability of a variable-precision routine for computing the first 2 N moments of the underlying weight function to any precision dig > nofdig. The procedure is applied to Freud, Bose–Einstein, and Fermi–Dirac orthogonal polynomials.

The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights

We consider orthogonal polynomials { p n , N ( x ) } n = 0 ∞ on the real line with respect to a weight w ( x ) = e − N V ( x ) and in particular the asymptotic behaviour of the coefficients a n , N and b n , N in the three-term recurrence x π n , N ( x ) = π n + 1 , N ( x ) + b n , N π n , N ( x ) + a n , N π n − 1 , N ( x ) . For one-cut regular V we show, using the Deift–Zhou method of steepest descent for Riemann–Hilbert problems, that the diagonal recurrence coefficients a n , n and b n , n have asymptotic expansions as n → ∞ in powers of 1 / n 2 and powers of 1 / n , respectively.

Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight

We study polynomials that are orthogonal with respect to the modified Laguerre weight z−n+νe−Nz(z− 1)2b, in the limit where n, N→∞ with N/n→ 1 and ν is a fixed number in . With the effect of the factor (z− 1)2b, the local parametrix near the critical point z= 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.

Partition function for multi-cut matrix models

We consider the partition function Z N of a random matrix model with polynomial potential V(ξ) = t 1 ξ + t 2 ξ 2 + ··· + t 2d ξ 2d . It is known that the second logarithmic derivative of Z N with respect to the times t k can be expressed in terms of the recurrence coefficients of the related orthogonal polynomials. An explicit formula for the recurrence coefficients of the orthogonal polynomials in the limit N → oo for multi-cut regular V (ξ) has been derived in [ 10] through the Riemann-Hilbert approach. The expression for Z N in the limit N → ∞ has been derived in [7] through a mean-field approach. We show that the above asymptotic formulae satisfy the same relations that hold for finite N.

Computing polynomials orthogonal with respect to densely oscillating and exponentially decaying weight functions and related integrals

Software (in Matlab) is developed for computing variable-precision recurrence coefficients for orthogonal polynomials with respect to the weight functions 1 + sin ( 1 / t ) , 1 + cos ( 1 / t ) , e - 1 / t on [ 0 , 1 ] , as well as e - 1 / t - t on [ 0 , ∞ ] and e - 1 / t 2 - t 2 on [ - ∞ , ∞ ] . Numerical examples are given involving Gauss quadrature relative to these weight functions.