Difference operators and difference equations on lattices, or grids, up to the elliptic hypergeometric case

TL;DR

This work develops a comprehensive framework for difference calculus on elliptic lattices defined by biquadratic curves, introducing a divided difference operator that preserves a controllable rational-degree growth: if f has degree d then Df attains degree 2d. The central construction uses elliptic lattices x_n=E(nh+t0) with E an elliptic function, enabling a two-pronged approach: (i) first- and second-order elliptic difference operators and equations, and (ii) interpolatory elliptic expansions and biorthogonal rational systems. The paper then develops an extensive theory of interpolants, Pell-type Pell-Lasserre identities, and the interplay with Padé approximants and continued fractions, culminating in second-order elliptic difference equations solved by hypergeometric-type and elliptic-hypergeometric expansions, and a rich adjoint-operator formalism. The results bridge theta-function formalisms and classical special-function theory, offering a unified algebraic-analytic setting for elliptic-difference equations with potential applications to elliptic hypergeometric systems and biorthogonal rational functions. The framework provides rigorous tools to analyze rational interpolants, orthogonality/biorthogonality relations, and spectral properties of elliptic-difference operators, with explicit connections to elliptic functions, theta-functions, and hypergeometric expansions.

Abstract

It is shown how to define difference operators and equations on particular lattices $\{x_n\}$, $2n\in\mathbb{Z}$, such that the divided difference operator $(\mathcal{D}f)(x_{n+1/2})= (f(x_{n+1})-f(x_n))/(x_{n+1}-x_n)$ has the property that $\mathcal{D}f$ is a rational function of degree $2d$ when $f$ is a rational function of degree $d$. It is then shown that the $x_n$s are in the most general case values of an elliptic function at a sequence of arguments in arithmetic progression (\emph{elliptic lattice}). Many special and limit cases, down to the most elementary ones, are considered too. First and second order difference operators and equations are constructed, up to the simplest elliptic hypergeometric ones. One also shows orthogonality and biorthogonality properties of rational solutions to some of these difference equations.

Figures (9)

• Figure 1: Examples of values involved in the solution of a difference equation. Why are the axes drawn without variables names?
• Figure 2: Some points involved in the solution of a difference equation.
• Figure 3: An instance of a biquadratic curve $F(x,y)=X_2(x)y^2+X_1(x)y+X_0(x)=0$, with $P(x)=42.26667(x+2)(x^2-1)(x-1.5), X_2(x)=(x+5.5)(x-4)$, and where $X_1(x)= -0.7333x^2 + 27.49x - 0.6882$ interpolates $\sqrt{P}$ at the zeros of $X_2$ (vertical asymptotes).
• Figure 4: Influence of $\epsilon$ on the step.
• Figure 5: A near fixed point and a true one.