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|Title:||Some important continued fractions of Ramanujan and Selberg|
|Doctoral Committee Chair(s):||Berndt, Bruce C.|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|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.
|Rights Information:||Copyright 1990 Zhang, Liang-Cheng|
|Date Available in IDEALS:||2011-05-07|
|Identifier in Online Catalog:||AAI9114482|
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