IDEALS Home University of Illinois at Urbana-Champaign logo The Alma Mater The Main Quad

Some important continued fractions of Ramanujan and Selberg

Show full item record

Bookmark or cite this item: http://hdl.handle.net/2142/19300

Files in this item

File Description Format
PDF 9114482.pdf (2MB) Restricted to U of Illinois (no description provided) PDF
Title: Some important continued fractions of Ramanujan and Selberg
Author(s): Zhang, Liang-Cheng
Doctoral Committee Chair(s): Berndt, Bruce C.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
Abstract: 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.We use contiguous relations for the generalized hypergeometric series $\sb3$F$\sb2$ to give new proofs of Ramanujan's elegant continued fractions for products and quotients of gamma functions. Previous proofs were somewhat ad hoc and did not show any connections with hypergeometric functions.
Issue Date: 1990
Type: Text
Language: English
URI: http://hdl.handle.net/2142/19300
Rights Information: Copyright 1990 Zhang, Liang-Cheng
Date Available in IDEALS: 2011-05-07
Identifier in Online Catalog: AAI9114482
OCLC Identifier: (UMI)AAI9114482
 

This item appears in the following Collection(s)

Show full item record

Item Statistics

  • Total Downloads: 1
  • Downloads this Month: 0
  • Downloads Today: 0

Browse

My Account

Information

Access Key