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A Specialised Continued Fraction

Published online by Cambridge University Press:  20 November 2018

A. J. Van Der Poorten  and
J. Shallit
A. J. Van Der Poorten*
Affiliation:
Centre for Number Theory Research Macquarie University NSW 2109 Australia
J. Shallit*
Affiliation:
Department of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1
*
email: alf@macadam.mpce.mq.edu.au
email: shallit®graceland.uwaterloo.ca
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Abstract

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We display a number with a surprising continued fraction expansion and show that we may explain that expansion as a specialisation of the continued fraction expansion of a formal series: A series ΣchX-h has a continued fraction expansion with partial quotients polynomials in X of positive degree (other, perhaps than the 0-th partial quotient). Simple arguments, let alone examples, demonstrate that it is noteworthy if those partial quotients happen to have rational integer coefficients only. In that special case one may replace the variable X by an integer ≥ 2; that is: one may 'specialise' and thereby proceed to obtain the regular continued fraction expansion of values of the series. And that is significant because, generally, it is difficult to obtain the explicit continued fraction expansion of a number presented in different shape. Our example leads to a series with a specialisable continued fraction expansion and, a little surprisingly, our arguments suggest that the phenomenon of specialisability for series of the kind appearing here may be reserved to just the special subclass of series we happen to have stumbled upon.

Keywords

11A5511Y6511J70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 5 , 01 October 1993 , pp. 1067 - 1079
Copyright
Copyright © Canadian Mathematical Society 1993

Footnotes

The first author was supported in part by grants from the Australian Research Council

The second author was supported in part by a grant from NSERC Canada

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