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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 ofshift 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 asystem of a very simple form. Using this we characterize functions satisfyingtwo linear scalar differential or difference equations with respect to theseoperators. We also indicate how these results have consequences both in thetheory of automatic sets, leading to a new proof of Cobham's Theorem, and inthe Galois theories of linear difference and differential equations, leading tohypertranscendence results.

Consistent systems of linear differential and difference equations