Consistent systems of linear differential and difference equations
Reinhard SchäfkeUniversité de Strasbourg, France
Michael F. SingerNorth Carolina State University, Raleigh, USA
We consider systems of linear differential and difference equations
with , a shift operator , -dilation operator or Mahler operator and systems of two linear difference equations
with a sufficiently independent pair of shift operators, pair of -dilation operators or pair of Mahler operators. Here and are matrices with rational function entries. Assuming a consistency hypothesis, we show that such systems 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.
Cite this article
Reinhard Schäfke, Michael F. Singer, Consistent systems of linear differential and difference equations. J. Eur. Math. Soc. 21 (2019), no. 9, pp. 2751–2792DOI 10.4171/JEMS/891