A recurrence relation generalizing those of Apéry

In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper, we explore those of the Racah and q-Racah types. For the M-indexed (q-)Racah polynomials, we derive 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 and annihilation operators of the quantum mechanical systems described by the multi-indexed (q-)Racah polynomials are obtained. In Appendix B and Appendix C, we present a proof and some data of the recurrence relations with constant coefficients for the multi-indexed Wilson and Askey-Wilson polynomials.

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szegő recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.

Orthogonal matrix polynomials and higher-order recurrence relations

A Generalization of Favard′s Theorem for Polynomials Satisfying a Recurrence Relation

The main difference between certain spectral problems for linear Schrödinger operators, e.g. the almost Mathieu equation, and three-term recurrence relations for orthogonal polynomials is that in the former the index ranges across ℤ and in the latter only across ℤ+. We present a technique that, by a mixture of Dirichlet and Taylor expansions, translates the almost Mathieu equation and its generalizations to three term recurrence relations. This opens up the possibility of exploiting the full power of the theory of orthogonal polynomials in the analysis of Schrödinger spectra.Aforementioned three-term recurrence relations share the property that their coefficients are almost periodic. In the special case when they form a periodic sequence, the support can be explicitly identified by a technique due to Geronimus. The more difficult problem, when the recurrence is almost periodic but fails to be periodic, is still open and we report partial results.The main promise of the technique of this article is that it can be extended to deal with multivariate extensions of the almost Mathieu equations. However, important theoretical questions need be answered before it can be implemented to the analysis of the spectral problem for Schrödinger operators.KeywordsOrthogonal PolynomialSpectral ProblemChebyshev PolynomialBorel MeasureEssential SpectrumThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

There are two definitions for the discrete q-Hermite polynomials, one of them is defined for \(0 1\). This paper is devoted to extend these definitions to the discrete q-Hermite matrix polynomials by means of the generating matrix functions. Explicit expressions and Rodrigues-type formulas for the discrete q-Hermite matrix polynomials are obtained. Some recurrence relations for these matrix polynomials, in particular the three terms recurrence relations are given. Furthermore, some identities are proved.

In this paper, we discuss the solution of singularly perturbed differential-difference equations exhibiting dual layer using the higher order finite differences. First, the second order singularly perturbed differential-difference equations is replaced by an asymptotically equivalent second order singular perturbed ordinary differential equation. Then, fourth order stable finite difference scheme is applied to get a three term recurrence relation which is easily solved by Thomas algorithm. Some numerical examples have been solved to validate the computational efficiency of the proposed numerical scheme. To analyze the effect of the parameters on the solution, the numerical solution has also been plotted using graphs. The error bound and convergence of the method have also been established.

An accurate Fortran code for computing hydrogenic continuum wave functions at a wide range of parameters

Relaxation dynamics of a particle in the presence of an external potential: exact solution in terms of matrix continued fractions

A stronger notion of nonnegative linearization of orthogonal polynomials is introduced. It requires that also the associated polynomials of any order have nonnegative linearization property. This turns out to be equivalent to a maximal principle of a discrete boundary value problem associated with orthogonal polynomials through the three term recurrence relation. The property is stable for certain perturbations of the recurrence relation. Criteria for the strong nonnegative linearization are derived. The range of parameters for the Jacobi polynomials satisfying this new property is determined.

Matrix polynomials satisfying first order differential equations and three term recurrence relations

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.

It is often desirable to obtain (asymptotic) properties of orthogonal polynomials and the measure with respect to which these polynomials are orthogonal. All orthogonal polynomials on the real line (with a positive Borel measure) satisfy a three term recurrence relation. We give a survey showing how properties of the recurrence coefficients reveal properties of the corresponding orthogonal polynomials.

The $J$-matrix method is extended to difference and $q$-difference operators and is applied to several explicit differential, difference, $q$-difference and second order Askey-Wilson type operators. The spectrum and the spectral measures are discussed in each case and the corresponding eigenfunction expansion is written down explicitly in most cases. In some cases we encounter new orthogonal polynomials with explicit three term recurrence relations where nothing is known about their explicit representations or orthogonality measures. Each model we analyze is a discrete quantum mechanical model in the sense of Odake and Sasaki [J. Phys. A: Math. Theor. 44 (2011), 353001, 47 pages].

2 × 2 hypergeometric operators with diagonal eigenvalues