{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/19300"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/19300","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Some important continued fractions of Ramanujan and Selberg","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.","abstract_html":"We provide explicit solutions for three q-difference equations which arise in Ramanujan and Selberg&#x27;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 <span class=\"etd-inline-math\">\\vert &gt;</span> 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 <span class=\"etd-inline-math\">\\vert &lt;</span> 1.","abstract_has_math":true,"creators":["Zhang, Liang-Cheng"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Berndt, Bruce C."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:03:12Z","date_published":"2011-05-07T12:03:12Z","updated_at":"2026-07-22T22:25:12Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 Zhang, Liang-Cheng"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114482","(UMI)AAI9114482"],"render_values":[{"text":"AAI9114482","href":null,"code":true},{"text":"(UMI)AAI9114482","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/19300","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Berndt, Bruce C."]},{"key":"dc:creator","label":"Author","values":["Zhang, Liang-Cheng"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:03:12Z","10000-01-01","1990"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1990 Zhang, Liang-Cheng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114482","(UMI)AAI9114482","http://hdl.handle.net/2142/19300"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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"]},{"key":"dc:title","label":"Title","values":["Some important continued fractions of Ramanujan and Selberg"]}]}],"canonical_facts":{"dc:contributor":["Berndt, Bruce C."],"dc:creator":["Zhang, Liang-Cheng"],"dc:date":["2011-05-07T12:03:12Z","10000-01-01","1990"],"dc:description":["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"],"dc:identifier":["AAI9114482","(UMI)AAI9114482","http://hdl.handle.net/2142/19300"],"dc:language":["eng"],"dc:rights":["Copyright 1990 Zhang, Liang-Cheng"],"dc:subject":["Mathematics"],"dc:title":["Some important continued fractions of Ramanujan and Selberg"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:12Z"}