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Generalizing the continued fraction algorithm to arbitrary dimensions
30th Annual Symposium on Foundations of Computer Science, 1989A new \(N\)-dimensional continued fraction algorithm is presented. Its most remarkable property is that it produces infinitely many solutions of the diophantine inequality \(\max_{1\leq i\leq N}| x_ i q-p_ i|\ll q^{-w(N)}\) with \(w(N)=1/2N(N+1)\). It also detects linear dependence.
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On some generalizations of Ramanujan’s continued fraction identities
Proceedings of the Indian Academy of Sciences - Section A, 1987Many of the continued fraction expansions of Ramanujan can be viewed as special cases of the expansion of the ratio of basic hypergeometric series: \(_ 2\phi_ 1(a,b;c;xq)/_ 2\phi_ 1(a,b;c;x).\) The authors find the related expansions for the ratios in which the numerator has been replaced by one of a number of similar basic hypergeometric series.
Bhargava, S. +2 more
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Other Generalizations of Continued Fractions
2013In this chapter we present some other generalizations of regular continued fractions to the multidimensional case. The main goal for us here is to give different geometric constructions related to such continued fractions (whenever possible). We say a few words about Minkowski–Voronoi continued fractions, triangle sequences related to Farey addition, O’
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Generalized continued fractions associated with the Gauss transform
Russian Mathematical Surveys, 2002In this short note the author describes some properties of the so-called \(\omega\)-continued fractions \(x=[a_0,a_1,\dots]_\omega\), which are a one-dimensional variant of the multidimensional \((A,\omega)\)-continued fractions, which was introduced by the author in [Math. Notes 56, No. 6, 1315--1317 (1994); translation from Mat. Zametki 56, No.
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Geometry of Continued Fractions
Algorithms and Computation in Mathematics, 2022Oleg N Karpenkov
exaly
Generalized Notions of Continued Fractions
2023Juan Fernández Sánchez +3 more
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Generation of Digital Planes Using Generalized Continued-Fractions Algorithms
Lecture Notes in Computer Science, 2016Xavier Provençal, Damien Jamet
exaly