Differential Transcendency of a Formal Laurent Series Satisfying a Rational Linear q-Difference Equation

Abstract

We prove that a formal Laurent series satisfying a rational linear q-difference equation of first order does not satisfy any nontrivial algebraic differential equation over the rational function field unless it represents a rational function. This also provides an algebraic proof of Ishizaki's theorem on differential transcendency for a meromorphic function satisfying a linear q-difference equation.