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ARITHMETIC PROPERTIES OF INFINITE PRODUCTS OF CYCLOTOMIC POLYNOMIALS

Part of: Diophantine approximation, transcendental number theory Differential and difference algebra

Published online by Cambridge University Press:  08 January 2016

PETER BUNDSCHUH  and
KEIJO VÄÄNÄNEN
PETER BUNDSCHUH*
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany email pb@math.uni-koeln.de
KEIJO VÄÄNÄNEN
Affiliation:
Department of Mathematical Sciences, University of Oulu, PO Box 3000, 90014 Oulu, Finland email keijo.vaananen@oulu.fi
*
✉pb@math.uni-koeln.de
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Abstract

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We study transcendence properties of certain infinite products of cyclotomic polynomials. In particular, we determine all cases in which the product is hypertranscendental. We then use various results from Mahler’s transcendence method to obtain algebraic independence results on such functions and their values.

Keywords

infinite productscyclotomic polynomialshypertranscendencealgebraic independenceMahler’s method

MSC classification

Primary: 11J91: Transcendence theory of other special functions
Secondary: 11J81: Transcendence (general theory) 12H05: Differential algebra 11J25: Diophantine inequalities
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 3 , June 2016 , pp. 375 - 387
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Amou, M., ‘Algebraic independence of the values of certain functions at a transcendental number’, Acta Arith. 59 (1991), 71–82.CrossRefGoogle Scholar
Bacher, R., ‘Twisting the Stern sequence’, Preprint, 2010, available at arXiv:1005.5627.Google Scholar
Bugeaud, Y., Han, G.-N., Wen, Z.-Y. and Yao, J.-Y., ‘Hankel determinants, Padé approximations, and irrationality exponents’, Preprint, 2015, available at arXiv:1503.02797.Google Scholar
Bundschuh, P., ‘Algebraic independence of infinite products and their derivatives’, in: Number Theory and Related Fields, Springer Proceedings in Mathematics and Statistics, 43 (eds. Borwein, J. M. et al. ) (Springer, New York, 2013), 143–156.CrossRefGoogle Scholar
Bundschuh, P. and Väänänen, K., ‘Algebraic independence of the generating functions of Stern’s sequence and of its twist’, J. Théor. Nombres Bordeaux 25 (2013), 43–57.CrossRefGoogle Scholar
Carlson, F., ‘Über Potenzreihen mit ganzzahligen Koeffizienten’, Math. Z. 9 (1921), 1–13.Google Scholar
Corvaja, P. and Zannier, U., ‘Some new applications of the subspace theorem’, Compositio Math. 131 (2002), 319–340.Google Scholar
Dreyfus, T., Hardouin, C. and Roques, J., ‘Hypertranscendence of solutions of Mahler equations’, Preprint, 2015, available at arXiv:1507.03361.Google Scholar
Duke, W. and Nguyen, H. N., ‘Infinite products of cyclotomic polynomials’, Bull. Aust. Math. Soc. 91 (2015), 400–411.CrossRefGoogle Scholar
Fatou, P., ‘Séries trigonométriques et séries de Taylor’, Acta Math. 30 (1906), 335–400.CrossRefGoogle Scholar
Kubota, K. K., ‘On the algebraic independence of holomorphic solutions of certain functional equations and their values’, Math. Ann. 227 (1977), 9–50.CrossRefGoogle Scholar
Nishioka, K., Mahler Functions and Transcendence, Lecture Notes in Mathematics, 1631 (Springer, Berlin, 1996).Google Scholar
Philippon, P., ‘Indépendance algébrique et K-fonctions’, J. reine angew. Math. 497 (1998), 1–15.CrossRefGoogle Scholar
Pólya, G. and Szegő, G., Aufgaben und Lehrsätze aus der Analysis II, Heidelberger Taschenbücher, 74 (Springer, Berlin, 1971).Google Scholar
Väänänen, K., ‘Algebraic independence of certain Mahler numbers’, Preprint, 2015, available at arXiv:1507.02510.Google Scholar
Waldschmidt, M., Diophantine Approximation on Linear Algebraic Groups, Grundlehren der Mathematischen Wissenschaften, 326 (Springer, Berlin, 2000).CrossRefGoogle Scholar