Some important continued fractions of Ramanujan and Selberg

We provide explicit solutions for three q-difference equations which arise in Ramanujan and Selberg's work on q-continued fractions. From these solutions, we derive criteria for the convergence of three Ramanujan-Selberg continued fractions when q is a primitive m-th root of unity. Moreover, when the continued fractions converge, we determine their values explicitly. For $\vert$ q \vert > 1, the continued fractions diverge, since the even and odd indexed convergents tend to distinct limits. We determine precisely these limits. We also give simple and uniform proofs of the three continued fraction formulas of Ramanujan and Selberg for $\vert$ q \vert < 1.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Zhang, Liang-Cheng

Contributors dc:contributor

• Berndt, Bruce C.

• Mathematics

Statement dc:rights

• Copyright 1990 Zhang, Liang-Cheng

Language dc:language

eng

Identifier

AAI9114482
(UMI)AAI9114482

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/19300