A recurrence formula for leaping convergents of non-regular continued fractions |
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Authors: | Carsten Elsner |
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Affiliation: | a Fachhochschule für die Wirtschaft, University of Applied Sciences, Freundallee 15, D-30173 Hannover, Germany b Faculty of Science and Technology, Hirosaki University, Hirosaki, 036-8561, Japan |
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Abstract: | Given a continued fraction [a0;a1,a2,…], pn/qn=[a0;a1,…,an] is called the n-th convergent for n=0,1,2,…. Leaping convergents are those of every r-th convergent prn+i/qrn+i (n=0,1,2,…) for fixed integers r and i with r?2 and i=0,1,…,r-1. This leaping step r can be chosen as the length of period in the continued fraction. Elsner studied the leaping convergents p3n+1/q3n+1 for the continued fraction of and obtained some arithmetic properties. Komatsu studied those p3n/q3n for (s?2). He has also extended such results for some more general continued fractions. Such concepts have been generalized in the case of regular continued fractions. In this paper leaping convergents in the non-regular continued fractions are considered so that a more general three term relation is satisfied. Moreover, the leaping step r need not necessarily to equal the length of period. As one of applications a new recurrence formula for leaping convergents of Apery’s continued fraction of ζ(3) is shown. |
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Keywords: | 05A19 11A55 11D04 11J70 15A06 |
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