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A Criterion for Irrationality

Published online by Cambridge University Press:  20 November 2018

R. F. Churchhouse
R. F. Churchhouse*
Affiliation:
Trinity Hall Cambridge, England
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The question of the irrationality of functions defined by power series, for rational values of the variable, has attracted much attention for over a hundred years. Legendre, in generalizing Lambert's proof of the irrationality of tan x for rational x, proved an important theorem on the irrationality of continued fractions with integer elements. Here we use Legendre's theorem (Lemma 3) to prove that at least one of a certain pair of power series is irrational whenever the variable is rational and satisfies a further condition. We prove the following:

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 253 - 260
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Bernstein, F. and Szász, O., Über Irrationalität unendlicher Kettenbrüche, Math. Ann., 76 (1915), 295–300.Google Scholar
2. Euler, L., Introductio in analysin infinitorum(Lausanne, 1748), 270.Google Scholar
3. Hall, H. S. and Knight, S. R., Higher algebra (London, 1882), 366.Google Scholar
4. Hardy, G. H. and Wright, E. M., Introduction to the theory of numbers (Oxford, 1938), 289.Google Scholar
5. Legendre, A. M., Géométrie et trigonométrie (Paris, 1821), 290.Google Scholar
6. Ramanujan, S., Collected papers (Cambridge, 1927), 214–215.Google Scholar
7. Rogers, L. J., Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. (1), 25 (1894), 329.Google Scholar