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

MR3308705 Hiroshi OGAWARA Hiroshi OGAWARA Differential Transcendency of a Formal Laurent Series Satisfying a Rational Linear $q$-Difference Equation Funkcialaj Ekvacioj. Serio Internacia 57 2014 477--488 http://fe.math.kobe-u.ac.jp/FE/FullPapers/57-3/57_477.pdf http://www.ams.org/mathscinet-getitem?mr=MR3308705 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. Differential transcendency, Differential algebra, Difference algebra, $q$-difference equations. 12H05, 12H10. 57-477 2014 Differential Transcendency of a Formal Laurent Series Satisfying a Rational Linear $q$-Difference Equation Hiroshi OGAWARA Hiroshi OGAWARA

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