Continued fractions with three limit points

George E. Andrews; Bruce C. Berndt; Jaebum Sohn; Ae Ja Yee; Alexandru Zaharescu

doi:10.1016/j.aim.2004.04.004

Continued fractions with three limit points

George E. Andrews, Bruce C. Berndt, Jaebum Sohn, Ae Ja Yee, Alexandru Zaharescu

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

On page 45 in his lost notebook, Ramanujan asserts that a certain q-continued fraction has three limit points. More precisely, if A_n/B_n denotes its nth partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of A_n/B_n exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established.

Original language	English (US)
Pages (from-to)	231-258
Number of pages	28
Journal	Advances in Mathematics
Volume	192
Issue number	2
DOIs	https://doi.org/10.1016/j.aim.2004.04.004
State	Published - Apr 1 2005

Keywords

Bauer-Muir transformation
Continued fraction
General convergence of continued fractions
Ramanujan's lost notebook
q-series

ASJC Scopus subject areas

General Mathematics

Online availability

10.1016/j.aim.2004.04.004

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title = "Continued fractions with three limit points",

abstract = "On page 45 in his lost notebook, Ramanujan asserts that a certain q-continued fraction has three limit points. More precisely, if An/Bn denotes its nth partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of An/Bn exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established.",

keywords = "Bauer-Muir transformation, Continued fraction, General convergence of continued fractions, Ramanujan's lost notebook, q-series",

author = "Andrews, {George E.} and Berndt, {Bruce C.} and Jaebum Sohn and Yee, {Ae Ja} and Alexandru Zaharescu",

note = "Funding Information: ·Corresponding author. Fax: +1-217-333-9576. E-mail addresses: [email protected] (G.E. Andrews), [email protected] (B.C. Berndt), [email protected] (J. Sohn), [email protected] (A.J. Yee), [email protected] (A. Zaharescu). 1Research partially supported by Grant DMS-9206993 from the National Science Foundation. 2Research partially supported by Grant MDA904-00-1-0015 from the National Security Agency. 3Research partially supported by the postdoctoral fellowship program from the Korea Science and Engineering Foundation, and by a grant from the Number Theory Foundation.",

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T1 - Continued fractions with three limit points

AU - Andrews, George E.

AU - Berndt, Bruce C.

AU - Sohn, Jaebum

AU - Yee, Ae Ja

AU - Zaharescu, Alexandru

N1 - Funding Information: ·Corresponding author. Fax: +1-217-333-9576. E-mail addresses: [email protected] (G.E. Andrews), [email protected] (B.C. Berndt), [email protected] (J. Sohn), [email protected] (A.J. Yee), [email protected] (A. Zaharescu). 1Research partially supported by Grant DMS-9206993 from the National Science Foundation. 2Research partially supported by Grant MDA904-00-1-0015 from the National Security Agency. 3Research partially supported by the postdoctoral fellowship program from the Korea Science and Engineering Foundation, and by a grant from the Number Theory Foundation.

PY - 2005/4/1

Y1 - 2005/4/1

N2 - On page 45 in his lost notebook, Ramanujan asserts that a certain q-continued fraction has three limit points. More precisely, if An/Bn denotes its nth partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of An/Bn exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established.

AB - On page 45 in his lost notebook, Ramanujan asserts that a certain q-continued fraction has three limit points. More precisely, if An/Bn denotes its nth partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of An/Bn exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established.

KW - Bauer-Muir transformation

KW - Continued fraction

KW - General convergence of continued fractions

KW - Ramanujan's lost notebook

KW - q-series

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JF - Advances in Mathematics

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Continued fractions with three limit points

Abstract

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ASJC Scopus subject areas

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