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

Department of Mathematics

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

7 Citations (Scopus)

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
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
Publication status	Published - 2005 Apr 1

Bibliographical note

Funding Information:
·Corresponding author. Fax: +1-217-333-9576. E-mail addresses: andrews@math.psu.edu (G.E. Andrews), berndt@math.uiuc.edu (B.C. Berndt), jsohn@yonsei.ac.kr (J. Sohn), yee@math.psu.edu (A.J. Yee), zaharesc@math.uiuc.edu (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.

All Science Journal Classification (ASJC) codes

Mathematics(all)

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10.1016/j.aim.2004.04.004

Cite this

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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.",

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: andrews@math.psu.edu (G.E. Andrews), berndt@math.uiuc.edu (B.C. Berndt), jsohn@yonsei.ac.kr (J. Sohn), yee@math.psu.edu (A.J. Yee), zaharesc@math.uiuc.edu (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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doi = "10.1016/j.aim.2004.04.004",

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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: andrews@math.psu.edu (G.E. Andrews), berndt@math.uiuc.edu (B.C. Berndt), jsohn@yonsei.ac.kr (J. Sohn), yee@math.psu.edu (A.J. Yee), zaharesc@math.uiuc.edu (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.

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ER -

Continued fractions with three limit points

Abstract

Bibliographical note

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