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

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.
Full text (pdf)
References ▼
  1. [1] J. Ax, On Schanuel's conjectures, Ann. of Math. 93 (1971), 252-268.
  2. [2] D.M. Bradley, Multiple \(q\)-zeta values, J. Algebra 283 (2005), 752-798.
  3. [3] R.M. Cohn, Difference Algebra, Interscience Publishers John Wiley & Sons, New York, London, Sydney, 1965.
  4. [4] G. Gasper, M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2nd ed., 2004.
  5. [5] A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique, IV, Publ. Math. Inst. Hautes Études Sci. 20 (1964), 5-259.
  6. [6] C. Hardouin, Hypertranscendance des systèmes aux différences diagonaux, Compos. Math. 144 (2008), 565-581.
  7. [7] J. Johnson, Kähler differentials and differential algebra, Ann. of Math. 89 (1969), 92-98.
  8. [8] V. Kac, P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002.
  9. [9] E.R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968), 1151-1164.
  10. [10] A. Levin, Difference Algebra, Algebra and Applications, vol. 8, Springer, New York, 2008.
  11. [11] T. Mansour, Identities for sums of a \(q\)-analogue of polylogarithm functions, Lett. Math. Phys. 87 (2009), 1-18.
  12. [12] M. Nishizawa, Generalized Hölder's theorem for Vignéras' multiple gamma function, Tokyo J. Math. 24 (2001), 323-329.
  13. [13] H. Ogawara, Another proof of Ostrowski-Kolchin-Hardouin theorem in difference algebra, Keio SFC Journal 13 (2013), 97-100.
  14. [14] A. Ostrowski, Sur les relations algébriques entre les intégrales indéfinies, Acta Math. 78 (1946), 315-318.
  15. [15] M. Rosenlicht, On Liouville's theory of elementary functions, Pacific J. Math. 65 (1976), 485-492.
  16. [16] M.F. Vignéras, L'équation fonctionnelle de la fonction zêta de Selberg du groupe modulaire \(\operatorname{PSL}(2,\mathbf{Z})\), [in:] Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), vol. 61 of Astérisque, Soc. Math. France, Paris, 1979, pp. 235-249.
  17. [17] J. Zhao, Multiple \(q\)-zeta functions and multiple \(q\)-polylogarithms, Ramanujan J. 14 (2007), 189-221.
About the authors ▼
Hiroshi Ogawara
Kumamoto University, Graduate School of Science and Technology, 2-39-1 Kurokami, Chuo-ku, Kumamoto, 860-8555, Japan
About the article ▼
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.
Download this article's citation as: