Classical Discrete Orthogonal Polynomials Research Articles

Sturm’s Comparison Theorem for Classical Discrete Orthogonal Polynomials

In an earlier work (Castillo et al. in J Math Phys 61:103505, 2020), it was established, from a hypergeometric-type difference equation, tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In this work, we continue with the study of zeros of these polynomials by giving a comparison theorem of Sturm type. As an application, we analyze in a simple way some relations between the zeros of certain classical discrete orthogonal polynomials.

Reduced-order state-space models for two-dimensional discrete systems via bivariate discrete orthogonal polynomials

This paper investigates the new model order reduction (MOR) methods via bivariate discrete orthogonal polynomials for two-dimensional (2-D) discrete systems. The 2-D discrete system is described by the Kurek model. First, we deduce algebraically the shift-transformation matrix of the classical discrete orthogonal polynomials of one variable. By means of the shift-transformation matrices, 2-D discrete systems are expanded in the spaces spanned by bivariate discrete orthogonal polynomials. The coefficient matrices are calculated from matrix equations. Then the reduced-order systems are produced by the orthogonal projection matrices defined by the coefficient matrices. Theoretical analysis shows that the reduced-order systems can match a certain number of coefficient vectors of the original outputs. Finally, one numerical example is simulated to demonstrate the feasibility and effectiveness of the proposed methods.

The electrostatic equilibrium problem for classical discrete orthogonal polynomials

We present two types of generic difference characterizations of zeros of the classical discrete orthogonal polynomials associated with a discrete weight function satisfying a difference equation of the Pearson type. These characterizations use either the total forward or modified partial central difference operators of discrete discriminants, which approximate the total electrostatic energy of the system of n n unit charges moving freely on an interval in presence of external potential given by the iterated discrete weight.

Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters

AbstractWe construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second‐order difference or differential operator. In mathematical physics, they allow the explicit computation of bound states of rational extensions of classical quantum‐mechanical potentials. The most apparent difference between classical or classical discrete orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials. The new examples have the novelty that they depend on an arbitrary number of continuous parameters.

Christoffel transform of classical discrete measures and invariance of determinants of classical and classical discrete polynomials

Given a finite set of complex numbers U and a classical discrete measure μ, we consider the Christoffel transform ∏u∈U(x−u)μ. Under mild conditions on the finite set U, we show that the orthogonal polynomials with respect to ∏u∈U(x−u)μ enjoy a bunch of non trivial determinantal representations in terms of the classical discrete orthogonal polynomials with respect to μ. Under additional assumptions on the finite set U, we dualize these determinantal representations and find some invariance properties for quasi Casoration determinants whose entries are classical discrete orthogonal polynomials. Passing to the limit we recover some known invariance properties for Wronskian determinant whose entries are classical polynomials.

Algorithms for Construction of Recurrence Relations for the Coefficients of the Fourier Series Expansions with Respect to Classical Discrete Orthogonal Polynomials

A new formula expressing explicitly the integrals, antidifference, of discrete orthogonal polynomials $$\{P_{n}(x):$$ Hahn, Meixner, Kravchuk, and Charlier $$\}$$ of any degree in terms of $$P_{n}(x)$$ themselves are proved. Other formulae for the expansion coefficients of general-order difference integrations $$\nabla ^{-s}f(x),\,\Delta ^{-s}f(x),$$ $$\nabla ^{-s}[x^{\ell }\nabla ^{q}f(x)]\,$$ and $$\Delta ^{-s}[x^{\ell }\Delta ^{q}f(x)],$$ of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained.

Stieltjes’ theorem for classical discrete orthogonal polynomials

The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, quantum q-Krawtchouk, q-Krawtchouk, affine q-Krawtchouk, q-Charlier, Al-Salam–Carlitz, q-Hahn, little q-Jacobi, little q-Laguerre/Wall, q-Bessel, q-Racah, and dual q-Hahn polynomials.

Invariance properties of Wronskian type determinants of classical and classical discrete orthogonal polynomials

Given a finite set F={f1,⋯,fk} of nonnegative integers (written in increasing order of magnitude) and a classical discrete family (pn)n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casorati determinant det⁡(pfi(x+j−1))i,j=1,⋯,k. In this paper we prove an invariance property of this kind of Casorati determinants when the set F is substituted by the set I(F)={0,1,2,⋯,max⁡F}∖{max⁡F−f:f∈F}. Our approach uses orthogonal polynomials that are eigenfunctions of higher order difference operators (Krall discrete polynomials). These polynomials are orthogonal with respect to certain Christoffel transforms of the classical discrete measures. By passing to the limit, this invariance property is extended to Wronskian type determinants whose entries are Hermite, Laguerre and Jacobi polynomials.

Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities

Let (pn)n be a sequence of orthogonal polynomials with respect to the measure μ. Let T be a linear operator acting in the linear space of polynomials P and satisfying deg(T(p))=deg(p)−1, for all polynomial p. We then construct a sequence of polynomials (sn)n, depending on T but not on μ, such that the Wronskian type n×n determinant det(Ti−1(pm+j−1(x)))i,j=1n is equal to the m×m determinant det(qn+i−1j−1(x))i,j=1m, up to multiplicative constants, where the polynomials qni, n,i⩾0, are defined by qni(x)=∑j=0nμjisn−j(x), and μji are certain generalized moments of the measure μ. For T=d/dx we recover a theorem by Leclerc which extends the well-known Karlin and Szegő identities for Hankel determinants whose entries are ultraspherical, Laguerre and Hermite polynomials. For T=Δ, the first order difference operator, we get some very elegant symmetries for Casorati determinants of classical discrete orthogonal polynomials. We also show that for certain operators T, the second determinant above can be rewritten in terms of Selberg type integrals, and that for certain operators T and certain families of orthogonal polynomials (pn)n, one (or both) of these determinants can also be rewritten as the constant term of certain multivariate Laurent expansions.

Symmetries for Casorati determinants of classical discrete orthogonal polynomials

Given a classical discrete family ( p n ) n (p_n)_n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn) and the set of numbers m + i − 1 m+i-1 , i = 1 , ⋯ , k i=1,\cdots ,k and k , m ≥ 0 k,m\ge 0 , we consider the k × k k\times k Casorati determinants det ( ( p n + j − 1 ( m + i − 1 ) ) i , j = 1 k ) \det ((p_{n+j-1}(m+i-1))_{i,j=1}^k) , n ≥ 0 n\ge 0 . In this paper, we conjecture a nice symmetry for these Casorati determinants and prove it for the cases k ≥ 0 , m = 0 , 1 k\ge 0, m=0,1 and m ≥ 0 , k = 0 , 1 m\ge 0, k=0,1 . This symmetry is related to the existence of higher order difference equations for the orthogonal polynomials with respect to certain Christoffel transforms of the classical discrete measures. Other symmetry will be conjectured for the Casorati determinants associated to the Meixner and Hahn families and the set of numbers − c + i -c+i , i = 1 , ⋯ , k i=1,\cdots ,k and k , m ≥ 0 k,m\ge 0 .

A basic class of symmetric orthogonal polynomials of a discrete variable

By using a generalization of Sturm–Liouville problems in discrete spaces, a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters, which generalizes all classical discrete symmetric orthogonal polynomials, is introduced. The standard properties of these polynomials, such as a second order difference equation, an explicit form for the polynomials, a three term recurrence relation and an orthogonality relation are presented. It is shown that two hypergeometric orthogonal sequences with 20 different weight functions can be extracted from this class. Moreover, moments corresponding to these weight functions can be explicitly computed. Finally, a particular example containing all classical discrete symmetric orthogonal polynomials is studied in detail.

Recurrence relation approach for expansion and connection coefficients in series of classical discrete orthogonal polynomials

The two formulae expressing explicitly the difference derivatives and the moments of discrete orthogonal polynomials {P n (x): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of P n (x) themselves are proved. Two other formulae for the expansion coefficients of general-order difference derivatives Δ q f(x), and for the moments x ℓΔ q f(x), of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Hahn–Charlier, Hahn–Meixner and Hahn–Kravchuk are described.

On zeros of discrete orthogonal polynomials

We exploit difference equations to establish sharp inequalities on the extreme zeros of the classical discrete orthogonal polynomials, Charlier, Krawtchouk, Meixner and Hahn. We also provide lower bounds on the minimal distance between their consecutive zeros.

Duplication coefficients via generating functions

In this article, we solve the duplication problem, where belongs to a wide class of polynomials, including the classical orthogonal polynomials (Hermite, Laguerre, Jacobi) as well as the classical discrete orthogonal polynomials (Charlier, Meixner, Krawtchouk) for the specific case a=-1. We give closed-form expressions as well as recurrence relations satisfied by the duplication coefficients.

Recurrences and explicit formulae for the expansion and connection coefficients in series of classical discrete orthogonal polynomials

Two formulae expressing explicitly the difference derivatives and the moments of a discrete orthogonal polynomials {P n (x): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of P n (x) themselves are proved. Two other formulae for the expansion coefficients of a general-order difference derivatives ∇ q f(x), and for the moments x ℓ∇ q f(x), of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica), in order to build and solve recursively for the connection coefficients between two families of Meixner, Kravchuk and Charlier, is described. Three analytical formulae for the connection coefficients between Hahn–Charlier, Hahn–Meixner and Hahn–Kravchuk are also developed.

Second order difference equations and discrete orthogonal polynomials of two variables

The second-order partial difference equation of two variables Du:=A1,1(x)Δ1∇1u+A1,2(x)Δ1∇2u+A2,1(x)Δ2∇1u+A2,2(x)Δ2∇2u+B1(x)Δ1u+B2(x)Δ2u=λu is studied to determine when it has orthogonal polynomials as solutions. We derive conditions on D so that a weight function W exists for which W D u is selfadjoint and the difference equation has polynomial solutions which are orthogonal with respect to W. The solutions are essentially the classical discrete orthogonal polynomials of two variables.

On Fourth-order Difference Equations for Orthogonal Polynomials of a Discrete Variable: Derivation, Factorization and Solutions

We derive and factorize the fourth-order difference equations satisfied by orthogonal polynomials obtained from some modifications of the recurrence coefficients of classical discrete orthogonal polynomials such as: the associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independent solutions of these fourth-order difference equations, and show how the results obtained for modified classical discrete orthogonal polynomials can be extended to modified semi-classical discrete orthogonal polynomials. Finally, we extend the validity of the results obtained for the associated classical discrete orthogonal polynomials with integer order of association from integers to reals.

Classical discrete orthogonal polynomials, Lah numbers, and involutory matrices

This letter proposes two different approaches in order to solve the connection problems P n(−ξ)= ∑ m=0 n C m(n)P m(ξ), when the family { P n ( x)} belongs to a wide class of polynomials, including classical discrete orthogonal polynomials. The first approach uses the involutory character of the Lah numbers and of other specific connection coefficients. The second approach presents the recursive Navima algorithm as applied to these problems.

DISCRETE ANALOGUES OF THE LAGUERRE INEQUALITY

It is shown that [Formula: see text], m = 0,1,…, where f(x) is either a real polynomial with only real zeros or an allied entire function of a special type, provided that the distance between two consecutive zeros of f(x) is at least [Formula: see text]. These inequalities are surprisingly similar discrete analogues of higher degree generalizations of the Laguerre and Turán inequalities. Applied to the classical discrete orthogonal polynomials, they yield sharp, explicit bounds, uniform in all parameters involved, on the polynomials and their extreme zeros. This is illustrated for the case of Krawtchouk polynomials.

Uniform Asymptotics for Polynomials Orthogonal with Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results

We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials become large. The class of orthogonal polynomials we consider includes as special cases the Krawtchouk and Hahn classical discrete orthogonal polynomials, but is far more general. In particular, we consider nodes that are not necessarily equally spaced. The asymptotic results are given with error bound for all points in the complex plane except for a finite union of discs of arbitrarily small but fixed radii. These exceptional discs are the neighborhoods of the so-called band edges of the associated equilibrium measure. As applications, we prove universality results for correlation functions of a general class of discrete orthogonal polynomial ensembles, and in particular we deduce asymptotic formulae with error bound for certain statistics relevant in the random tiling of a hexagon with rhombus-shaped tiles. The discrete orthogonal polynomials are characterized in terms of a a Riemann-Hilbert problem formulated for a meromorphic matrix with certain pole conditions. By extending the methods of [17, 22], we suggest a general and unifying approach to handle Riemann-Hilbert problems in the situation when poles of the unknown matrix are accumulating on some set in the asymptotic limit of interest.