University of Illinois Urbana-Champaign

Some important continued fractions of Ramanujan and Selberg

Zhang, Liang-Cheng

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https://hdl.handle.net/2142/19300

Description

Title
Some important continued fractions of Ramanujan and Selberg
Author(s)
Zhang, Liang-Cheng
Issue Date
1990
Doctoral Committee Chair(s)
Berndt, Bruce C.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
Type of Resource
text
Permalink
http://hdl.handle.net/2142/19300
Copyright and License Information
Copyright 1990 Zhang, Liang-Cheng

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