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On certain Mahler functions

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Abstract

A few years ago, Pellarin introduced a new sequence of functions each one being holomorphic in the unit disc and satisfying a Mahler-type functional equation. It will be shown that every single function is hypertranscendental. Moreover, a certain subsequence consisting of functions that are algebraically independent over \({\mathbb{C}(z)}\) is exhibited, and all other functions are \({\mathbb{Q}}\)-linear combinations of these modulo \({\mathbb{Q}[z]}\). Mahler’s method allows to deduce the corresponding arithmetic analogs from these analytic facts.

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References

  1. Bundschuh P.: Transcendence and algebraic independence of series related to Stern’s sequence. Int. J. Number Theory, 8, 361–376 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Carlson F.: Über Potenzreihen mit ganzen Koeffizienten, Math. Z., 9, 1–13 (1921)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kubota K.K.: On the algebraic independence of holomorphic solutions of certain functional equations and their values. Math. Ann., 227, 9–50 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  4. K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann., 101 (1929), 342--366; Berichtigung, ibid., 103 (1930), 532.

  5. Nishioka K.: A note on differentially algebraic solutions of first order linear difference equations. Aequationes Math., 27, 32–48 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. K. Nishioka, Mahler Functions and Transcendence, LNM 1631, Springer (Berlin et al., 1996).

  7. F. Pellarin, An introduction to Mahler’s method for transcendence and algebraic independence, arXiv:1005.1216v2 [math.NT] (2011).

  8. T. Töpfer, Algebraic independence of the values of generalized Mahler functions,Acta Arith., 70 (1995), 161--181.

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Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Köln, Germany

    P. Bundschuh

  2. Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014, Oulu, Finland

    K. Väänänen

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  1. P. Bundschuh
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  2. K. Väänänen
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Correspondence to P. Bundschuh.

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Bundschuh, P., Väänänen, K. On certain Mahler functions. Acta Math. Hungar. 145, 150–158 (2015). https://doi.org/10.1007/s10474-014-0451-z

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  • DOI: https://doi.org/10.1007/s10474-014-0451-z

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