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ON THE TRANSCENDENCE OF CERTAIN REAL NUMBERS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  20 February 2019

VEEKESH KUMAR Open the ORCID record for VEEKESH KUMAR [Opens in a new window]  and
BILL MANCE Open the ORCID record for BILL MANCE [Opens in a new window]
VEEKESH KUMAR
Affiliation:
Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad – 211019, India email veekeshkumar@hri.res.in
BILL MANCE*
Affiliation:
Uniwersytet im. Adama Mickiewicza w Poznaniu, Collegium Mathematicum, ul. Umultowska 87, 61-614 Poznań, Poland email william.mance@amu.edu.pl
*
✉william.mance@amu.edu.pl
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Abstract

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In this article, we prove the transcendence of certain infinite sums and products by applying the subspace theorem. In particular, we extend the results of Hančl and Rucki [‘The transcendence of certain infinite series’, Rocky Mountain J. Math.35 (2005), 531–537].

Keywords

transcendencerational approximationsubspace theorem

MSC classification

Primary: 11J68: Approximation to algebraic numbers
Secondary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 392 - 402
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author was supported by a research grant from the Department of Atomic Energy, Government of India.

References

Adamczewski, B., Bugeaud, Y. and Luca, F., ‘Sur la complexité des nombres algébriques’, C. R. Math. Acad. Sci. Paris 339(1) (2004), 11–14.Google Scholar
Adhikari, S. D., Saradha, N., Shorey, T. N. and Tijdeman, R., ‘Transcendental infinite sums’, Indag. Math. (N.S.) 12 (2001), 1–14.Google Scholar
Erdős, P. and Straus, E. G., ‘On the irrationality of certain series’, Pacific J. Math. 55 (1974), 85–92.Google Scholar
Hančl, J., ‘Two criteria for transcendental sequences’, Le Matematiche 56 (2001), 149–160.Google Scholar
Hančl, J., ‘A criterion for linear independence of series’, Rocky Mountain J. Math. 34 (2004), 173–186.Google Scholar
Hančl, J., Kolouch, O. and Novotný, L., ‘A criterion for linear independence of infinite products’, An. Ştiinţ. Univ. “Ovidius” Constanta Ser. Mat. 32 (2015), 107–120.Google Scholar
Hančl, J. and Rucki, P., ‘The transcendence of certain infinite series’, Rocky Mountain J. Math. 35 (2005), 531–537.Google Scholar
Nishioka, K., Mahler Functions and Transcendence, Lecture Notes in Mathematics, 1631 (Springer, New York, 1996).Google Scholar
Nyblom, M. A., ‘A theorem on transcendence of infinite series’, Rocky Mountain J. Math. 30 (2001), 1111–1120.Google Scholar
Nyblom, M. A., ‘A theorem on transcendence of infinite series II’, J. Number Theory 91 (2001), 71–80.Google Scholar
Schmidt, W. M., Diophantine Approximations and Diophantine Equations, Lecture Notes in Mathematics, 1467 (Springer, Berlin, 1991).Google Scholar