General convergence of continued fractions

Jacobsen, Lisa

doi:10.1090/s0002-9947-1986-0825716-1

Cited by 36 publications

(11 citation statements)

References 5 publications

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“…General convergence. In [7], Jacobsen revolutionised the subject of the convergence of continued fractions by introducing the concept of general convergence. General convergence is defined, see [8], as follows.…”

Section: If Y = Exp(2πit) Then R(y) Has Subsequences Of Approximantsmentioning

confidence: 99%

“…Jacobsen shows in [7] that, if a continued fraction converges in the general sense, then the limit is unique. The idea of general convergence is of great significance because classical convergence implies general convergence (take v n = 0 and w n = ∞, for all n), but the converse does not necessarily hold.…”

Section: If Y = Exp(2πit) Then R(y) Has Subsequences Of Approximantsmentioning

confidence: 99%

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On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

Bowman

McLaughlin

2003

Trans. Amer. Math. Soc.

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Abstract. This paper studies ordinary and general convergence of the RogersRamanujan continued fraction.Let the continued fraction expansion of any irrational number t ∈ (0, 1) be denoted by [0, e 1 (t), e 2 (t), · · · ] and let the i-th convergent of this continued fraction expansion be denoted bywhere φ = (It is shown that if y ∈ Y S , then the Rogers-Ramanujan continued fraction R(y) diverges at y. S is an uncountable set of measure zero. It is also shown that there is an uncountable set of points G ⊂ Y S such that if y ∈ G, then R(y) does not converge generally.It is further shown that R(y) does not converge generally for |y| > 1. However we show that R(y) does converge generally if y is a primitive 5m-th root of unity, for some m ∈ N. Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.

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Section: If Y = Exp(2πit) Then R(y) Has Subsequences Of Approximantsmentioning

confidence: 99%

Section: If Y = Exp(2πit) Then R(y) Has Subsequences Of Approximantsmentioning

confidence: 99%

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

Bowman

McLaughlin

2003

Trans. Amer. Math. Soc.

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“…In [2] a more general concept of convergence was introduced: we require that there exist two sequences {u n } and {v n } from C such that…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…If (1.7) holds, we say that K(a n /1) converges generally to c. Then, by [2], there exists an exceptional sequence…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Continued fractions with circular twin value sets

Lorentzen

2008

Trans. Amer. Math. Soc.

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“…We are often interested in sequences F n for which F n ( j) converges to a point on the ideal boundary C ∞ (in the chordal metric). Such sequences are said to be generally convergent; they were first studied in the context of continued fractions by Lorentzen in [8].…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

The Parabola Theorem on Continued Fractions

Short

2016

Comput. Methods Funct. Theory

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Using geometric methods borrowed from the theory of Kleinian groups, we interpret the parabola theorem on continued fractions in terms of sequences of Möbius transformations. This geometric approach allows us to relate the Stern-Stolz series, which features in the parabola theorem, to the dynamics of certain sequences of Möbius transformations acting on three-dimensional hyperbolic space. We also obtain a version of the parabola theorem in several dimensions.

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General convergence of continued fractions

Abstract: ABSTRACT. We introduce a new concept of convergence of continued fractions-general convergence. Moreover, we compare it to the ordinary convergence concept and to strong convergence. Finally, we prove some properties of general convergence.

Cited by 36 publications

References 5 publications

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

Continued fractions with circular twin value sets

The Parabola Theorem on Continued Fractions

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