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

Zbl 1104.00017

RatDiff; Maple
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