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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• Hiroshi Ogawara
• Kumamoto University, Graduate School of Science and Technology, 2-39-1 Kurokami, Chuo-ku, Kumamoto, 860-8555, Japan
• Communicated by P.A. Cojuhari.
• Received: 2016-05-30.
• Revised: 2016-10-19.
• Accepted: 2016-10-27.
• Published online: 2017-01-30.

Cite this article as:
Hiroshi Ogawara, On criteria for algebraic independence of collections of functions satisfying algebraic difference relations, Opuscula Math. 37, no. 3 (2017), 457-472, http://dx.doi.org/10.7494/OpMath.2017.37.3.457

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