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Title of article :
A recurrence formula for leaping convergents of non-regular continued fractions Original Research Article
Author/Authors :
Carsten Elsner، نويسنده , , Takao Komatsu، نويسنده ,
Issue Information :
روزنامه با شماره پیاپی سال 2008
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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 rgreater-or-equal, slanted2 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 image and obtained some arithmetic properties. Komatsu studied those p3n/q3n for image (sgreater-or-equal, slanted2). 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.
Keywords :
Leaping convergents , continued fractions , Linear recurrence relations
Journal title :
Linear Algebra and its Applications
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Link To Document :