Hyperexponential solutions of finite-rank ideals in orthogonal ore rings

Abstract

Abstract. An orthogonal Ore algebra is an abstraction of common properties of linear partial differential, shift and q-shift operators. Using orthogonal Ore algebras, we present an algorithm for finding hyperexponential solutions of a system of linear differential, shift and q-shift operators, or any mixture thereof, whose solution space is finite-dimensional. The algorithm is applicable to factoring modules over an orthogonal Ore algebra when the modules are also finite-dimensional vector spaces over the field of rational functions.

Picard--Vessiot extensions for linear functional systems

“…The first statement is proved. The second is immediate from (6). From Theorem 1, there exist an orthogonal ∆-extension E of F and a fundamental matrix U with entries in E for D.…”

“…Let (F, Φ) be a ∆-field, and suppose that for each i such that σi = 1, there exists ai ∈ F such that σi(ai) = ai and σj(ai) − ai = δj(ai) = 0 for all j = i. Replacing the xi by the ai in the proof of Theorem 1 in [6], one sees that linear functional equations over F can be rewritten as equations over an orthogonal ∆-field. There are however orthogonal ∆-rings that do not contain such ai's, for example F = C(x) together with Φ = {(1, d/dx), (σx, 0)} where σx is the automorphism of F over C that sends x to x − 1.…”

“…There are however orthogonal ∆-rings that do not contain such ai's, for example F = C(x) together with Φ = {(1, d/dx), (σx, 0)} where σx is the automorphism of F over C that sends x to x − 1. This field is used in modeling differential-delay equations, and does not match the definition of orthogonality given in [6].…”

“…Note that the conditions (2) are derived from the condition ∂i(∂j(Z)) = ∂j(∂i(Z)) and are exactly the matrix-analogues of the compatibility conditions for first order scalar equations in [6]. Example 1.…”

Determining Whether a Multivariate Hyperexponential Function is Algebraic

“…Hyperexponential functions in several variables are an abstraction of common properties of exponential functions and hypergeometric terms (see [6]). They play important roles in factoring modules over Laurent-Ore algebras, and in the continuous and discrete versions of Zeilberger's algorithm for proving combinatorial identities (see [11] and [7], respectively).…”