Opuscula Math. 37, no. 3 (2017), 457-472

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.
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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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