Consistent systems of linear differential and difference equations

TL;DR

This paper studies systems of linear differential and difference equations, showing they can be simplified under certain conditions, and explores implications for automatic sets, Cobham's theorem, and Galois theories.

Contribution

It provides a reduction method for consistent systems of linear differential and difference equations and characterizes functions satisfying two such equations, with applications to automatic sets and Galois theories.

Findings

Systems can be reduced to simple forms under consistency.

Characterization of functions satisfying two linear equations.

Connections established with automatic sets and Galois theories.

Abstract

We consider systems of linear differential and difference equations \begin{eqnarray*} \partial Y(x) =A(x)Y(x), \sigma Y(x) =B(x)Y(x) \end{eqnarray*} with $\partial = \frac{d}{dx}$, $\sigma$ a shift operator $\sigma(x) = x+a$, $q$-dilation operator $\sigma(x) = qx$ or Mahler operator $\sigma(x) = x^p$ and systems of two linear difference equations \begin{eqnarray*} \sigma_1 Y(x) =A(x)Y(x), \sigma_2 Y(x) =B(x)Y(x) \end{eqnarray*} with $(\sigma_1,\sigma_2)$ a sufficiently independent pair of shift operators, pair of $q$-dilation operators or pair of Mahler operators. Here $A(x)$ and $B(x)$ are $n\times n$ matrices with rational function entries. Assuming a consistency hypothesis, we show that such system can be reduced to a system of a very simple form. Using this we characterize functions satisfying two linear scalar differential or difference equations with respect to these operators. We also indicate how these results have consequences both in the theory of automatic sets, leading to a new proof of Cobham's Theorem, and in the Galois theories of linear difference and differential equations, leading to hypertranscendence results.