[Paper Review] Second order difference equations and discrete orthogonal polynomials of two variables

Yuan Xu|ArXiv.org|Jul 27, 2004

Quantum Mechanics and Non-Hermitian Physics6 references3 citations

TL;DR

This paper characterizes second-order partial difference equations in two variables that admit discrete orthogonal polynomials as solutions by establishing conditions under which the difference operator is self-adjoint with respect to a weight function W. The key result identifies eight classes of such equations, corresponding to classical discrete orthogonal polynomials of two variables—Hahn, Meixner, Krawtchouk, Charlier, and their product forms—on lattice sets like the simplex and rectangle, under mild structural constraints on the coefficient functions.

ABSTRACT

The second order partial difference equation of two variables $ \CD u:= A_{1,1}(x) Δ_1 abla_1 u + A_{1,2}(x) Δ_1 abla_2 u + A_{2,1}(x) Δ_2 abla_1 u + A_{2,2}(x) Δ_2 abla_2 u & \qquad \qquad \qquad \qquad + B_1(x) Δ_1 u + B_2(x) Δ_2 u = λu, $ is studied to determine when it has orthogonal polynomials as solutions. We derive conditions on $\CD$ so that a weight function $W$ exists for which $W \CD u$ is self-adjoint 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.

Motivation & Objective

• To determine the conditions under which a second-order partial difference equation in two variables admits orthogonal polynomial solutions.
• To identify the class of difference operators for which a weight function W exists such that W𝒟 is self-adjoint.
• To establish a consistency condition between the weight function W and the difference operator 𝒟 to ensure polynomial solutions are orthogonal.
• To classify all such equations that yield classical discrete orthogonal polynomials in two variables, including product-type and Hahn-type systems.
• To show that under mild structural assumptions on the coefficient functions, these are the only possible equations with orthogonal polynomial solutions.

Proposed method

• Define the second-order partial difference operator 𝒟 using forward and backward difference operators Δi and ∇i on a lattice in ℤ².
• Impose the condition that the mixed second-order coefficients satisfy A₁₂ − A₂₁ ∈ Π₁² (linear polynomials), ensuring symmetry in the quadratic part.
• Derive necessary and sufficient conditions for the existence of a weight function W such that W𝒟 is formally self-adjoint with respect to the discrete inner product.
• Introduce the notion of W-consistency with the difference equation, ensuring that the weight vanishes on the boundary of the domain in a way compatible with the operator.
• Use the self-adjointness and consistency conditions to prove that polynomial solutions exist and are orthogonal with respect to W.
• Classify the resulting equations by analyzing the coefficient polynomials Aᵢⱼ and Bᵢ, leading to the identification of eight canonical families of orthogonal polynomials.

Experimental results

Research questions

• RQ1 Under what conditions on the coefficient functions Aᵢⱼ and Bᵢ does the second-order partial difference equation 𝒟u = λu have orthogonal polynomial solutions?
• RQ2 When does a weight function W exist such that the operator W𝒟 is self-adjoint with respect to the discrete inner product?
• RQ3 What is the precise relationship between the weight function W and the difference operator 𝒟 that ensures the orthogonality of polynomial solutions?
• RQ4 Which classical discrete orthogonal polynomials in two variables arise as solutions to such equations, and on which lattice domains do they live?
• RQ5 Are there any other second-order difference equations beyond the identified families that can support orthogonal polynomial solutions under reasonable assumptions?

Key findings

• The paper identifies eight distinct classes of second-order partial difference equations in two variables that admit orthogonal polynomial solutions.
• These solutions correspond to classical discrete orthogonal polynomials: Hahn polynomials on the simplex V = {x₁ ≥ 0, x₂ ≥ 0, x₁ + x₂ ≤ N}, and product-type polynomials (Meixner, Krawtchouk, Charlier, and their mixed forms) on rectangular domains V = {0 ≤ x₁ ≤ M, 0 ≤ x₂ ≤ N}.
• For each such equation, a weight function W is explicitly constructed such that the operator W𝒟 is self-adjoint and the polynomial solutions are orthogonal with respect to W.
• The conditions for self-adjointness and W-consistency are both necessary and sufficient for the existence of orthogonal polynomial solutions.
• The analysis shows that under the assumption A₁₂ − A₂₁ ∈ Π₁² (linear polynomials), no other nontrivial difference equations yield orthogonal polynomial solutions.
• The discrete Hahn polynomials on the simplex are shown to be the discrete analog of Jacobi-type polynomials on the triangle, but no discrete analog exists for the orthogonal polynomials on the disk.