{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86775"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86775","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Generalizations of Certain Results on Continued Fraction","abstract":"In this thesis we study generalizations of the Rogers-Ramanujan continued fraction. The Rogers-Ramanujan continued fraction arises from a three-term q-difference equation. We consider (m + 1)-term q-difference equations and also a generalization of the continued fraction algorithm called a G-continued fraction. We obtain a general expansion of the quotient of two contiguous basic hypergeometric function in arbitrarily many variables as a G-continued fraction. A careful interpretation of convergence is given for different cases of this expansion. When a full vector space of solutions of a q-difference equation is known, we use the theorem of Zahar which extends a theorem of Pincherle. When this is not the case, we apply the theory on infinite system of equations to the G-continued fraction in order to obtain convergence. Also, an explicit formula for the approximants of a G-continued fraction is given. An application of this formula is used to obtain a combinatorial interpretation of a G-continued fraction extension of the Rogers-Ramanujan continued fraction. A combinatorial interpretation of the coefficients of the q-difference equation for a very well-poised basic hypergeometric series studied by A. Selberg is derived. Finally, the arithmetic properties of a generalization of the Rogers-Ramanujan continued fraction are considered.","abstract_html":"In this thesis we study generalizations of the Rogers-Ramanujan continued fraction. The Rogers-Ramanujan continued fraction arises from a three-term q-difference equation. We consider (m + 1)-term q-difference equations and also a generalization of the continued fraction algorithm called a G-continued fraction. We obtain a general expansion of the quotient of two contiguous basic hypergeometric function in arbitrarily many variables as a G-continued fraction. A careful interpretation of convergence is given for different cases of this expansion. When a full vector space of solutions of a q-difference equation is known, we use the theorem of Zahar which extends a theorem of Pincherle. When this is not the case, we apply the theory on infinite system of equations to the G-continued fraction in order to obtain convergence. Also, an explicit formula for the approximants of a G-continued fraction is given. An application of this formula is used to obtain a combinatorial interpretation of a G-continued fraction extension of the Rogers-Ramanujan continued fraction. A combinatorial interpretation of the coefficients of the q-difference equation for a very well-poised basic hypergeometric series studied by A. Selberg is derived. Finally, the arithmetic properties of a generalization of the Rogers-Ramanujan continued fraction are considered.","abstract_has_math":false,"creators":["Choi, Geumlan"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Douglas Bowman"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:28Z","date_published":"2015-09-28T15:19:28Z","updated_at":"2026-07-22T22:26:27Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3017047"],"render_values":[{"text":"(MiAaPQ)AAI3017047","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86775","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Douglas Bowman"]},{"key":"dc:creator","label":"Author","values":["Choi, Geumlan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:28Z","10000-01-01","2001"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86775","(MiAaPQ)AAI3017047"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis we study generalizations of the Rogers-Ramanujan continued fraction. The Rogers-Ramanujan continued fraction arises from a three-term q-difference equation. We consider (m + 1)-term q-difference equations and also a generalization of the continued fraction algorithm called a G-continued fraction. We obtain a general expansion of the quotient of two contiguous basic hypergeometric function in arbitrarily many variables as a G-continued fraction. A careful interpretation of convergence is given for different cases of this expansion. When a full vector space of solutions of a q-difference equation is known, we use the theorem of Zahar which extends a theorem of Pincherle. When this is not the case, we apply the theory on infinite system of equations to the G-continued fraction in order to obtain convergence. Also, an explicit formula for the approximants of a G-continued fraction is given. An application of this formula is used to obtain a combinatorial interpretation of a G-continued fraction extension of the Rogers-Ramanujan continued fraction. A combinatorial interpretation of the coefficients of the q-difference equation for a very well-poised basic hypergeometric series studied by A. Selberg is derived. Finally, the arithmetic properties of a generalization of the Rogers-Ramanujan continued fraction are considered.","Made available in DSpace on 2015-09-28T15:19:28Z (GMT). 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A combinatorial interpretation of the coefficients of the q-difference equation for a very well-poised basic hypergeometric series studied by A. Selberg is derived. Finally, the arithmetic properties of a generalization of the Rogers-Ramanujan continued fraction are considered.","Made available in DSpace on 2015-09-28T15:19:28Z (GMT). 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