Hardouin, Charlotte; Singer, Michael F. Differential Galois theory of linear difference equations. (English) Zbl 1163.12002 Math. Ann. 342, No. 2, 333-377 (2008). 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)]. Reviewer: Jacques Sauloy (Toulouse) Cited in 9 ReviewsCited in 73 Documents 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 Keywords:difference equations; differential Galois theory Citations:Zbl 1104.00017 Software:Maple; RatDiff × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Abramov, S.A.: The rational component of the solution of a first order linear recurrence relation with rational right hand side. Ž. Vyčisl. Mat. i Mat. Fiz. 15(4), 1035–1039, 1090 (1975) · Zbl 0326.65069 [2] Abramov, S.A., Zima, E.V.: D’Alembert solutions of inhomogeneous linear equations (differential, difference and otherwise). In: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, pp. 232–239. 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