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Berndt, Bruce C. Zhang, Liang-Cheng 2011-05-07T12:03:12Z 2011-05-07T12:03:12Z 10000-01-01 1990 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. Made available in DSpace on 2011-05-07T12:03:12Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114482.pdf: 2455612 bytes, checksum: cedfb48ce003af290b1d1e6ea10a564b (MD5) Previous issue date: 1990 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:36:01Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:14:25-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9114482 (UMI)AAI9114482 http://hdl.handle.net/2142/19300 eng Copyright 1990 Zhang, Liang-Cheng Mathematics Some important continued fractions of Ramanujan and Selberg text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.