Term Recurrence Relation Research Articles

A Matlab package computing simultaneous Gaussian quadrature rules for multiple orthogonal polynomials

The aim of this paper is to describe a Matlab package for computing the simultaneous Gaussian quadrature rules associated with a variety of multiple orthogonal polynomials.Multiple orthogonal polynomials can be considered as a generalization of classical orthogonal polynomials, satisfying orthogonality constraints with respect to r different measures, with r≥1. Moreover, they satisfy (r+2)-term recurrence relations. In this manuscript, without loss of generality, r is considered equal to 2. The so-called simultaneous Gaussian quadrature rules associated with multiple orthogonal polynomials can be computed by solving a banded lower Hessenberg eigenvalue problem. Unfortunately, computing the eigendecomposition of such a matrix turns out to be strongly ill-conditioned and the Matlab function balance.m does not improve the condition of the eigenvalue problem. Therefore, most procedures for computing simultaneous Gaussian quadrature rules are implemented with variable precision arithmetic. Here, we propose a Matlab package that allows to reliably compute the simultaneous Gaussian quadrature rules in floating point arithmetic. It makes use of a variant of a new balancing procedure, recently developed by the authors of the present manuscript, that drastically reduces the condition of the Hessenberg eigenvalue problem.

Para-Bannai-Ito Polynomials

New bispectral polynomials orthogonal on a Bannai-Ito bi-lattice (uniform quadri-lattice) are obtained from an unconventional truncation of the untruncated Bannai-Ito and complementary Bannai-Ito polynomials. A complete characterization of the resulting para-Bannai-Ito polynomials is provided, including a three term recurrence relation, a Dunkl-difference equation, an explicit expression in terms of hypergeometric series and an orthogonality relation. They are also derived as a $q\to -1$ limit of the $q$-para-Racah polynomials. A connection to the dual $-1$ Hahn polynomials is also established.

Matrix Jacobi biorthogonal polynomials via Riemann–Hilbert problem

We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann–Hilbert problem we can derive first and second order differential-difference relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlevé d-P I V _{IV} equations are obtained for the three term recurrence relation coefficients.

Bounds for Extreme Zeros of Classical Orthogonal Polynomials Related to Birth and Death Processes

In this paper, we consider birth and death processes with different sequences of transition rates and find the bound for the extreme zeros of orthogonal polynomials related to the three term recurrence relations and birth and death processes. Furthermore, we find the related chain sequences. Using these chain sequences, we find the transition probabilities for the corresponding process. As a consequence, transition probabilities related to G-fractions and modular forms are derived. Results obtained in this work are new and several graphical representations and numerical computations are provided to validate the results.

Spectral properties related to generalized complementary Romanovski–Routh polynomials

Complementary Romanovski–Routh polynomials play an important role in extracting specific properties of orthogonal polynomials. In this work, a generalized form of the Complementary Romanovski–Routh polynomials (GCRR) that has the Gaussian hypergeometric representation and satisfies a particular type of recurrence called $$R_{II}$$ type three term recurrence relation involving two arbitrary parameters is considered. Self perturbation of GCRR polynomials leading to extracting two different types of $$R_{II}$$ type orthogonal polynomials are identified. Spectral properties of these resultant polynomials in terms of tri-diagonal linear pencil are analyzed. The LU decomposition of these pencil matrices provided interesting properties involving biorthogonality. Interlacing properties between the zeros of the polynomials in the discussion are established.

On Perfectness of Systems of Weights Satisfying Pearson’s Equation with Nonstandard Parameters

Measures generating classical orthogonal polynomials are determined by Pearson’s equation, whose parameters usually provide the positivity of the measures. The case of general complex parameters (nonstandard) is also of interest; the non-Hermitian orthogonality with respect to (now complex-valued) measures is considered on curves in C. Some applications lead to multiple orthogonality with respect to a number of such measures. For a system of r orthogonality measures, the perfectness is an important property: in particular, it implies the uniqueness for the whole family of corresponding multiple orthogonal polynomials and the (r+2)-term recurrence relations. In this paper, we introduce a unified approach which allows to prove the perfectness of the systems of complex measures satisfying Pearson’s equation with nonstandard parameters. We also study the polynomials satisfying multiple orthogonality relations with respect to a system of discrete measures. The well-studied families of multiple Charlier, Krawtchouk, Meixner and Hahn polynomials correspond to the systems of measures defined by the difference Pearson’s equation with standard real parameters. Using the same approach, we verify the perfectness of such systems for general parameters. For some values of the parameters, discrete measures should be replaced with the continuous measures with non-real supports.

A generalized quaternionic sequence with Vietoris' number components

In this investigation, the aim is to determine a generalized quaternionic sequence with Vietoris' number components depending on 2-parameters ? and ?. Considering specific real values ? and ?, various types of classical quaternionic sequence with Vietoris' number components can be obtained as real, split, split-semi and so on. The fundamental algebraic structures, several classical expressions, a two and three term recurrence relations are identified, as well as Catalan-like, generating function and Binet-like formulas. Furthermore, a determinantal approach is used to generate the generalized quaternionic sequence with Vietoris' number components.

Study of an example of Markov chains involving Chebyshev polynomials

In his paper, ‘Random walks and orthogonal polynomials: some challenges’, F. A. Grunbaum gave the polynomials orthogonal with respect to the weight on explicitly as where and are, respectively, the Chebyshev polynomials of the first and second types. In this paper, similarly, we introduce the polynomials defined by where and are, respectively, the Chebyshev polynomials of the third and fourth types, then we give the three term recurrence relation of the polynomials . Second, we give the kernel where is the Christoffel–Darboux formula for the polynomials . Finally we give the integral of function of and we show how we deduce that is orthogonal with respect to the weight on .

On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case

In this contribution we consider sequences of monic polynomials orthogonal with respect to the Sobolev-type inner product f,g=⟨uM,fg⟩+λTjf(α)Tjg(α), where uM is the Meixner linear operator, λ∈R+, j∈N, α≤0, and T is the forward difference operator Δ or the backward difference operator ∇. Moreover, we derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of the second order is also given. In addition, for these polynomials, we derive a (2j+3)-term recurrence relation. Finally, we find the Mehler–Heine type formula for the particular case α=0.

Associated orthogonal polynomials of the first kind and Darboux transformations

Let u be a quasi-definite linear functional defined on the linear space of polynomials P with complex coefficients. For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) (Pn)n∈N, which satisfies a three term recurrence relation. If we increase the indices of the recurrence relation by one unity, then we get the sequence of associated polynomials of the first kind as solution. These polynomials will be orthogonal with respect to a linear functional denoted by u(1). In the literature two special transformations of the functional u are studied, the canonical Christoffel transformation u˜=(x−c)u and the canonical Geronimus transformation uˆ=(x−c)−1u+Mδc, where c is a fixed complex number, M is a free complex parameter and δc is the linear functional defined on P as 〈δc,p(x)〉=p(c). For the Christoffel transformation with SMOP (P˜n)n∈N, we are interested in analyzing the relation between the linear functionals u(1) and u˜(1). The super index denotes the linear functionals associated with the orthogonal polynomial sequences of the first kind (Pn(1))n∈N and (P˜n(1))n∈N, respectively. This problem is also studied for Geronimus transformations. Here we give close relations between their corresponding monic Jacobi matrices by using the LU and UL factorizations. To get this result, we first need to study the relation between u−1 (the inverse functional) and u(1) which can be given from a quadratic Geronimus transformation.

Space-time fractional diffusion equation associated with Jacobi expansions

We consider the operator , where is the three term recurrence relation for the Jacobi polynomial normalized at the point x = 1 and I is the identity operator acting on . Knowing that for , these polynomials satisfy a product formula which provides a hypergroup structure on , we use the tools of harmonic analysis on this hypergroup to solve the space-time fractional diffusion equation Here, , is the Caputo time fractional derivative of order σ. The solution is given by , being the fundamental solution expressed in terms of Mittag-Leffler function and is the convolution on the hypergroup . We prove that, for , is nonnegative and we give some of its properties. Next, we study the one parameter operators associated with this fundamental solution.

Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices

In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as $$1/k_i$$ , $$k_i\in {\mathbb {Z}}_{>0}$$ ( $$i=1,\,2$$ ), the average characteristic polynomials admit $$(k_1+k_2+2)$$ -term recurrence relations, which can be interpreted as spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest solvable example is given.

Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials

The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as a more encompassing characterization of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree ℓD≥1, the three term recurrence relations are broken. The extra ℓD “lower degree polynomials,” which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree ℓD+N. The discrete orthogonality relations are shown to hold for another group of “new” orthogonal polynomials called Krein–Adler polynomials based on the Hermite, Laguerre, and Jacobi polynomials.

On a quaternionic sequence with Vietoris’ numbers

Special integers sequences have been the center of attention for many researchers, as well as the sequences of quaternions where its components are the elements of these sequences. Motivated by a rational sequence, we consider the quaternions with components Vietoris? numbers and investigate some of its properties. For this sequence a two and three term recurrence relation is established, as well as a Binet?s type formula. Moreover the generating function for this sequence is introduced and also the determinant of some tridiagonal matrices are used in order to find elements of this sequence.

Bivariate Poly-analytic Hermite Polynomials

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier–Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical Hilbert space on the two-complex space with respect to the Gaussian measure. Their basic properties are discussed, such as their three term recurrence relations, operational realizations and differential equations (Bochner’s property) they obey. Different generating functions of exponential type are obtained. Integral and exponential operational representations are also derived. Some applications in the context of integral transforms and the concrete spectral theory of specific magnetic Laplacians are discussed.

On the Darboux transformations and sequences of p-orthogonal polynomials

For a fixed $p \in \mathbb{N}$, sequences of polynomials $\{P_n\}$, $n \in \mathbb{N}$, defined by a $(p+2)$-term recurrence relation are related to several topics in Approximation Theory. A $(p+2)$-banded matrix $J$ determines the coefficients of the recurrence relation of any of such sequences of polynomials. The connection between these polynomials and the concept of orthogonality has been already established through a $p$-dimension vector of functionals. This work goes further in this topic by analyzing the relation between such vectors for the set of sequences $\{P_n^{(j)}\}$, $n \in N$, associated with the Darboux transformations $J^{(j)}$, $j=1, ..., p,$ of a given $(p+2)$-banded matrix $J$.

Recurrence relations of the multi-indexed orthogonal polynomials. VI. Meixner–Pollaczek and continuous Hahn types

In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, Askey–Wilson, Racah, and q-Racah types. In this paper, we explore those of the Meixner–Pollaczek and continuous Hahn types. For the M-indexed Meixner–Pollaczek and continuous Hahn polynomials, we present 3 + 2M term recurrence relations with variable dependent coefficients and 1 + 2L term (L ≥ M + 1) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation/annihilation operators of the quantum mechanical systems described by the multi-indexed Meixner–Pollaczek and continuous Hahn polynomials are obtained.

Ratio asymptotics for biorthogonal matrix polynomials with unbounded recurrence coefficients

In this paper we study matrix biorthogonal polynomials sequences that satisfy a nonsymmetric three term recurrence relation with unbounded matrix coefficients. The outer ratio asymptotics for this family of matrix biorthogonal polynomials is derived under quite general assumptions. Some illustrative examples are considered.

2 × 2 hypergeometric operators with diagonal eigenvalues

In this work we give all the order-two hypergeometric operators D, symmetric with respect to some 2 × 2 irreducible matrix-weight W on (0,1) such that DPn=Pnλn00μn with no repetition among the eigenvalues {λn,μn}n∈N0, where {Pn}n∈N0 is the (unique) sequence of monic orthogonal polynomials with respect to W.We obtain a new family of such operators and weights, depending on three parameters, generalizing some older examples. We also give, in a very explicit way, the corresponding monic orthogonal polynomials, their three term recurrence relation and their squared matrix-norms.

Application of a complete radiation boundary condition for the Helmholtz equation in locally perturbed waveguides

This paper deals with an application of a complete radiation boundary condition (CRBC) for the Helmholtz equation in locally perturbed waveguides. The CRBC, one of efficient high-order absorbing boundary conditions, has been analyzed in straight waveguides in Hagstrom and Kim (2019). In this paper, we apply CRBC to the Helmholtz equation posed in locally perturbed waveguides and establish the well-posedness of the problem and convergence of CRBC approximate solutions. The new CRBC proposed in this paper improves the one studied in Hagstrom and Kim (2019) in two aspects. The first one is that the new CRBC involves more damping parameters with the same computational cost as that of CRBC in Hagstrom and Kim (2019), which results in 50% smaller reflection errors. The second one is that the new CRBC takes a Neumann terminal condition of three term recurrence relations of auxiliary variables instead of a Dirichlet terminal condition used in Hagstrom and Kim (2019) so that it can treat cutoff modes effectively. Finally, we present numerical experiments illustrating the convergence theory.