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On the irrationality of certain series

Published online by Cambridge University Press:  24 October 2008

Peter B. Borwein
Peter B. Borwein
Affiliation:
Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada
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Abstract

We prove that the series

are irrational and not Liouville whenever q is an integer (q ╪ 0, ±1) and r is a nonzero rational (r ╪ −qn).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 1 , July 1992 , pp. 141 - 146
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Borwein, J. M. and Borwein, P. B.. Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity (Wiley, 1987).Google Scholar
[2]Borwein, P. B.. On the irrationality of ∑1/(q n+r). J. Number Theory 37 (1991), 253–259.CrossRefGoogle Scholar
[3]Borwein, P. B.. Padé approximants for the q-elementary functions. Constr. Approx. 4 (1988), 391–402.CrossRefGoogle Scholar
[4]Chudnovsky, D. V. and Chudnovsky, G. V.. Padé and rational approximation to systems of functions and their arithmetic applications. In Number Theory, New York 1982, Lecture Notes in Math. vol. 1052 (Springer-Verlag, 1984).CrossRefGoogle Scholar
[5]Erdös, P.. On arithmetical properties of Lambert Series. J. Indian Math. Soc. (N.S.) 12 (1948), 63–66.Google Scholar
[6]Erdös, P.. On the irrationality of certain series: problems and results. In New Advances in Transcendence Theory (Cambridge University Press, 1988), pp. 102–109.Google Scholar
[7]Erdös, P. and Graham, R. L.. Old and New Problems and Results in Combinatorial Number Theory. Enseign. Math. Monograph no. 28 (1980).Google Scholar
[8]Gasper, G. and Rahman, M.. Basic Hypergeometric Series (Cambridge University Press, 1990).Google Scholar
[9]Mahler, K.. Zur Approximation der Expontialfunkion und des Logarithmus. J. Reine Angew. Math. 166 (1931), 118–150.Google Scholar
[10]Matala-Aho, T.. Remarks on the arithmetic properties of certain hypergeometric series of Gauss and Heine. Ph.D. thesis, Oulu University (1991).Google Scholar
[11]Wallisser, R.. Rational Approximation des q-Analogons der Exponential-funktion und Irrationalitätsaussagen für diese Funktion. Arch. Math. (Basel) 44 (1985), 59–64.CrossRefGoogle Scholar