Consistent systems of linear differential and difference equations

Abstract

We consider systems of linear differential and difference equationswith δ = d dx , σ a shift operator σ(x) = x + a, q-dilation operator σ(x) = qx or Mahler operator σ(x) = x p and systems of two linear difference equationswith (σ 1 , σ 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 × n 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.

Galois groups for integrable and projectively integrable linear difference equations

“…They prove the validity of their criteria by appealing to Bézivin's result [3], which states that a power series that simultaneously satisfies a Mahler equation and a linear differential equation must be a rational function, and to Ramis's result [21], which states that a power series that simultaneously satisfies a q-difference equation and a linear differential equation must also be a rational function. These classical results of Bézivin and Ramis are reproved and generalized by Schäfke and Singer in [23], where they also classify series and other functions that simultaneously satisfy either a shift-difference equation and a linear differential equation, or a pair of difference equations for suitable pairs of operators. The authors of [23] prove these results by considering systems of linear differential and difference equations…”

“…These classical results of Bézivin and Ramis are reproved and generalized by Schäfke and Singer in [23], where they also classify series and other functions that simultaneously satisfy either a shift-difference equation and a linear differential equation, or a pair of difference equations for suitable pairs of operators. The authors of [23] prove these results by considering systems of linear differential and difference equations…”

“…where A(x) ∈ GL n (C 0 (x)), C 0 is an algebraically closed field, and σ is a shift operator, qdilation operator, or Mahler operator. Using Theorem 1 of [23], we characterize those groups that occur as differential Galois groups under the assumption that (1.1) is integrable (i.e., its solutions also satisfy a linear differential system) or projectively integrable 1 (i.e., it becomes integrable after "moding out by scalars"; for precise definitions see Definitions 3.2 and 4.1).…”

“…The rest of the paper is organized as follows. A key idea of this paper is that the results of [23] allow one to reduce questions concerning the differential Galois groups of integrable and projectively integrable equations (1.1) to questions concerning difference systems with constant coefficients. In Section 2 we discuss systems with constant coefficients and their Galois groups.…”

“…Using the Galois theory developed in [18], they develop Galois-theoretic criteria to deduce σ q ′ -transcendence results. Difference systems with such pairs of operators were also considered in [23]. Theorem 12 of this latter paper should yield new proofs and extensions of some of these results, but we do not consider this case or other pairs of difference operators in this paper.…”