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Differential Galois theory of linear difference equations. (English) Zbl 1163.12002
The authors prove hypertranscendency properties for solutions of linear difference and \(q\)-difference equations (like the Gamma function, a result due to Hölder); and, more generally, properties of differential-algebraic independence for families of such solutions. This is in the line of previous work by the first author. For this, they develop a Galois theory of difference and \(q\)-difference equations based on differential algebraic groups and on parameterized differential Galois theory, as studied in a previous work by the second author and P. Cassidy [Differential equations and quantum groups. IRMA Lect. Math. Theor. Phys. 9, 113–155 (2007; Zbl 1104.00017)].

MSC:
12H05 Differential algebra
39A10 Additive difference equations
33B15 Gamma, beta and polygamma functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
39A12 Discrete version of topics in analysis
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