Three-term Recurrence Relation Research Articles

On the spectrum of tridiagonal operators in the context of orthogonal polynomials

AbstractThe basis for our studies is a large class of orthogonal polynomial sequences $$(P_n)_{n\in {{\mathbb {N}}}_0}$$ ( P n ) n ∈ N 0 , which is normalized by $$P_n(x_0)=1$$ P n ( x 0 ) = 1 for all $$n\in {\mathbb {N}}_0$$ n ∈ N 0 where the coefficients in the three-term recurrence relation are bounded. The goal is to check if $$x_0 \in {\mathbb {R}}$$ x 0 ∈ R is in the support of the orthogonalization measure $$\mu $$ μ . For this purpose, we use, among other things, a result of G. H. Hardy concerning Cesàro operators on weighted $$l^2$$ l 2 -spaces. These investigations generalize ideas from Lasser et al. (Arch Math 100:289–299, 2013).

Recognition of 3D Images by Fusing Fractional-Order Chebyshev Moments and Deep Neural Networks.

In order to achieve efficient recognition of 3D images and reduce the complexity of network parameters, we proposed a novel 3D image recognition method combining deep neural networks with fractional-order Chebyshev moments. Firstly, the fractional-order Chebyshev moment (FrCM) unit, consisting of Chebyshev moments and the three-term recurrence relation method, is calculated separately using successive integrals. Next, moment invariants based on fractional order and Chebyshev moments are utilized to achieve invariants for image scaling, rotation, and translation. This design aims to enhance computational efficiency. Finally, the fused network embedding the FrCM unit (FrCMs-DNNs) extracts depth features to analyze the effectiveness from the aspects of parameter quantity, computing resources, and identification capability. Meanwhile, the Princeton Shape Benchmark dataset and medical images dataset are used for experimental validation. Compared with other deep neural networks, FrCMs-DNNs has the highest accuracy in image recognition and classification. We used two evaluation indices, mean square error (MSE) and peak signal-to-noise ratio (PSNR), to measure the reconstruction quality of FrCMs after 3D image reconstruction. The accuracy of the FrCMs-DNNs model in 3D object recognition was assessed through an ablation experiment, considering the four evaluation indices of accuracy, precision, recall rate, and F1-score.

The solution for singularly perturbed differential-difference equation with boundary layers at both ends by a numerical integration method

This paper presents a numerical integration method for the solution of ‘Singularly Perturbed Differential-Difference Equations' having dual layers. It is well known that when we use already existing numerical methods to solve such problems, we get oscillatory or unsatisfactory results unless we take a very small step size, which is time-consuming and costly. To get a numerical solution for such a problem, first, the delay and advanced parameters present in the SPDDE are approximated by Taylor’s series to get an equivalent ‘Singularly Perturbed Differential Equation' of second order. Second, an asymptotically equivalent first-order differential equation is obtained from SPDE using Taylor’s transformation. Composite Simpson’s 1/3 rule is implemented to get a three-term recurrence relation. The Thomas algorithm is applied to get the solution of the tri-diagonal system of equations. Several model examples are tested and it was found that the numerical solution approximates the available/exact solution very well.

QBD Processes Associated with Jacobi–Koornwinder Bivariate Polynomials and Urn Models

We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi–Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.

On computing modified moments for half-range Hermite weights

AbstractIn this paper, we consider the computation of the modified moments for the system of Laguerre polynomials on the real semiaxis with the Hermite weight. These moments can be used for the computation of integrals with the Hermite weight on the real semiaxis via product rules. We propose a new computational method based on the construction of the null-space of a rectangular matrix derived from the three-term recurrence relation of the system of orthonormal Laguerre polynomials. It is shown that the proposed algorithm computes the modified moments with high relative accuracy and linear complexity. Numerical examples illustrate the effectiveness of the proposed method.

Gaussian type quadrature rules related to the oscillatory modification of the generalized Laguerre weight functions

In this paper we consider weighted integrals with respect to a modification of the generalized Laguerre weight functions wα(x)=xαe−x, α>−1, on (0,+∞) by the oscillatory factor exp(iζx), where ζ>0. For calculation of such integrals for analytic real-valued functions in D={z∈ℂ∣Rez≥0,Imz≥0} we construct polynomials orthogonal with respect to the linear functional L:P→ℂ, given by L[p]=∫0+∞p(x)wα(x)exp(iζx)dx, where P is a linear space of all algebraic polynomials, as well as the corresponding quadrature rules of Gaussian type. The existence of such orthogonal polynomials and the corresponding three-term recurrence relations are proved. A few numerical examples are presented, including computation of the Cauchy principal value integrals.

Truncated Hermite polynomials

We define the family of truncated Hermite polynomials , orthogonal with respect to the linear functional The connection of with Hermite and Rys polynomials is stated. The semiclassical character of as polynomials of class 2 is emphasized. As a consequence, several properties of concerning the coefficients in the three-term recurrence relation they satisfy as well as the moments and the Stieltjes function of L are given. Ladder operators associated with such a linear functional and the holonomic equation that the polynomials satisfy are deduced.

Recurrence and Eigenfunction Methods for Non-Trivial Models of Discrete Binary Choice.

Understanding how systems relax to equilibrium is a core theme of statistical physics, especially in economics, where systems are known to be subject to extrinsic noise not included in simple agent-based models. In models of binary choice-ones not much more complicated than Kirman's model of ant recruitment-such relaxation dynamics become difficult to determine analytically and require solving a three-term recurrence relation in the eigendecomposition of the stochastic process. In this paper, we derive a concise closed-form solution to this linear three-term recurrence relation. Its solution has traditionally relied on cumbersome continued fractions, and we instead employ a linear algebraic approach that leverages the properties of lower-triangular and tridiagonal matrices to express the terms in the recurrence relation using a finite set of orthogonal polynomials. We pay special attention to the power series coefficients of Heun functions, which are also important in fields such as quantum mechanics and general relativity, as well as the binary choice models studied here. We then apply the solution to find equations describing the relaxation to steady-state behavior in social choice models through eigendecomposition. This application showcases the potential of our solution as an off-the-shelf solution to the recurrence that has not previously been reported, allowing for the easy identification of the eigenspectra of one-dimensional, one-step, continuous-time Markov processes.

Discrete harmonic analysis associated with Jacobi expansions III: The Littlewood–Paley–Stein [formula omitted]-functions and the Laplace type multipliers

The research about harmonic analysis associated with Jacobi expansions carried out in Arenas et al. (2020) and Arenas et al. (2022) is continued in this paper. Given the operator J(α,β)=J(α,β)−I, where J(α,β) is the three-term recurrence relation for the normalized Jacobi polynomials and I is the identity operator, we define the corresponding Littlewood–Paley–Stein gk(α,β)-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.

The bilateral birth–death chain generated by the associated Jacobi polynomials

AbstractWe give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three‐term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one‐step transition probability matrix of a discrete‐time bilateral birth–death chain with state space on . We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.

A Stieltjes Algorithm for Generating Multivariate Orthogonal Polynomials

Orthogonal polynomials of several variables have a vector-valued three-term recurrence relation, much like the corresponding one-dimensional relation. This relation requires only knowledge of certain recurrence matrices, and allows simple and stable evaluation of multivariate orthogonal polynomials. In the univariate case, various algorithms can stably and accurately evaluate the recurrence coefficients given the ability to compute polynomial moments, but it is difficult to identify analogous procedures in multiple dimensions. We present a new Multivariate Stieltjes (MS) algorithm that fills this gap in the multivariate case, allowing computation of recurrence matrices assuming moments are available. The algorithm is essentially explicit in two and three dimensions, but requires the numerical solution to a nonconvex problem in more than three dimensions. Compared to direct Gram–Schmidt-type orthogonalization, we demonstrate on several examples in up to three dimensions that the MS algorithm is far more stable, and allows accurate computation of orthogonal bases in the multivariate setting, in contrast to direct orthogonalization approaches.

Asymptotics of matrix valued orthogonal polynomials on [−1,1

We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann–Hilbert formulation for MVOPs and the Deift–Zhou method of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree tends to infinity, in different regions of the complex plane (outside the interval of orthogonality, on the interval away from the endpoints and in neighborhoods of the endpoints), as well as for the matrix coefficients in the three-term recurrence relation for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires several different factorizations of the matrix part of the weight, in terms of eigenvalues/eigenvectors and using a matrix Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.

Classical solutions of the degenerate fifth Painlevé equation

In this paper classical solutions of the degenerate fifth Painlevé equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlevé equation are known to be expressible in terms of solutions of the third Painlevé equation. The classification and description of the classical solutions of the degenerate fifth Painlevé equation is done using the Hamiltonian associated with third Painlevé equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.

An algebraic approach for the Dunkl–Killingbeck problem from the bi-confluent Heun equation

In this paper, we study the Dunkl–Killingbeck problem in two dimensions. We apply the Lie algebraic approach within the framework of quasi-exact solvability to the radial part of the Dunkl–Killingbeck problem to find the general exact expressions for the energies and corresponding wave functions. The allowed values of the potential parameters are the representation space of sl(2) Lie algebra. In addition, we discuss that the effective potential of the Dunkl–Killingbeck is the same as the obtained from the bi-confluent Heun equation by a suitable variable transformation. Following earlier results, we follow the explicit solutions of this differential equation expressed as a series expansion of Hermite functions and obtain the expansion coefficients from a three-term recurrence relation. In the sequel, we present that this construction leads to the known quasi-exactly solvable (QES) form of the Dunkl–Killingbeck problem. Therefore, we find that the expressions for the energy eigenvalues and wave functions of the corresponding potential term are in agreement with those from the QES formalism. Then, we derive the ladder operators for the Dunkl–Killingbeck problem within the algebraic approach. It seems that this method is the Dunkl–Killingbeck rotation problem solved by operators of the su[Formula: see text] Lie algebra in a specific way.

Two-dimensional contiguous relations for the linearization coefficients of classical orthogonal polynomials

ABSTRACT By using the three-term recurrence relation for orthogonal polynomials, we produce a collection of two-dimensional contiguous relations for certain generalized hypergeometric functions. These generalized hypergeometric functions arise through linearization coefficients for some classical orthogonal polynomials in the Askey-scheme, namely Gegenbauer (ultraspherical), Hermite, Jacobi and Laguerre polynomials.

Orthogonal polynomials for self-dual weights

A class of orthogonal polynomials relative to special discrete weights is considered. These self-dual weights which are completely determined by their finite support appear in the polynomial approximation of a function over this set, in the barycentric form of classical Lagrange interpolation polynomial as well as in mathematical statistics and statistical physics. Orthogonal polynomials possess remarkable symmetry reflected in the properties of their coefficients in the three-term recurrence relations, some of which have an explicit form. Formulas for the expected values are derived.

Computational Approach for a Singularly Perturbed Differential Equations With Mixed Shifts Using a Non-Polynomial Spline

In this paper, a second order singularly perturbed differential difference equation with both the negative and positive shifts is considered. A fitted non-polynomial spline approach is applied to solve the problem. Taylor series expansion process is being used to produce an approximated form of the considered problem, and then a fitted non-polynomial spline approach is devised in the form of a three-term recurrence relation. The convergence of the method is examined, and a quadratic rate of convergence is achieved. The maximum absolute error with quadratic rate of convergence of the solution is recorded. Layer profile is examined using the graphs.

Novel Formulas of Schröder Polynomials and Their Related Numbers

This paper explores the Schröder polynomials, a class of polynomials that produce the famous Schröder numbers when x=1. The three-term recurrence relation and the inversion formula of these polynomials are a couple of the fundamental Schröder polynomial characteristics that are given. The derivatives of the moments of Schröder polynomials are given. From this formula, the moments of these polynomials and also their high-order derivatives are deduced as two significant special cases. The derivatives of Schröder polynomials are further expressed in new forms using other polynomials. Connection formulas between Schröder polynomials and a few other polynomials are provided as a direct result of these formulas. Furthermore, new expressions that link some celebrated numbers with Schröder numbers are also given. The formula for the repeated integrals of these polynomials is derived in terms of Schröder polynomials. Furthermore, some linearization formulas involving Schröder polynomials are established.

Principal Solutions of Recurrence Relations and Irrationality Questions in Number Theory

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of linear three-term recurrence relations, we show that there exists a four-term linear recurrence relation whose solutions show that the number has an irrational square if and only if the four-term recurrence relation has a principal solution of a certain type. The result is extended to higher-order recurrence relations, and a transcendence criterion can also be formulated in terms of these principal solutions. The method generates new series expansions of positive integer powers of ζ(3) and ζ(2) in terms of Apéry’s now classic sequences.

Revisiting recurrence criteria of birth and death processes. Short proofs

The paper contains several new transparent proofs of criteria appearing in classification of birth and death processes (BDPs). They are almost purely probabilistic and differ from the classical techniques of three-term recurrence relations, continued fractions and orthogonal polynomials. Let T ∞ {T^\infty } be the passage time from zero to ∞ \infty . The regularity criterion says that T ∞ > ∞ {T^\infty } > \infty if and only if E T ∞ > ∞ \mathbb {E}{T^\infty } > \infty . It is heavily based on a result of Gong, Y., Mao, Y.-H. and Zhang, C. [J. Theoret. Probab. 25 (2012), no. 4, 950–980]. We obtain the latter expectation by using a two-term recurrence relation. We observe that the recurrence criterion is an immediate consequence of the well-known recurrence criterion for discrete-time BDPs and a result of Chung K. L. [Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York (1967)]. We obtain the classical criterion of positive recurrence using technique of the common probability space. While doing so, we construct a monotone sequence of BDPs with finite state spaces converging to BDPs with an infinite state space.