Opuscula Math. 37, no. 3 (2017), 457-472
http://dx.doi.org/10.7494/OpMath.2017.37.3.457

Opuscula Mathematica

# On criteria for algebraic independence of collections of functions satisfying algebraic difference relations

Hiroshi Ogawara

Abstract. This paper gives conditions for algebraic independence of a collection of functions satisfying a certain kind of algebraic difference relations. As applications, we show algebraic independence of two collections of special functions: (1) Vignéras' multiple gamma functions and derivatives of the gamma function, (2) the logarithmic function, $$q$$-exponential functions and $$q$$-polylogarithm functions. In a similar way, we give a generalization of Ostrowski's theorem.

Keywords: difference algebra, systems of algebraic difference equations, algebraic independence, Vignéras' multiple gamma functions, $$q$$-polylogarithm functions.

Mathematics Subject Classification: 12H10, 39A10, 39A13.

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