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Title: | Modular identities for the Rogers-Ramanujan functions and analogues |
Author(s): | Gugg, Chadwick |
Director of Research: | Berndt, Bruce C. |
Doctoral Committee Chair(s): | Ahlgren, Scott |
Doctoral Committee Member(s): | Berndt, Bruce C.; Ford, Kevin; Stolarsky, Kenneth B. |
Department / Program: | Mathematics |
Discipline: | Mathematics |
Degree Granting Institution: | University of Illinois at Urbana-Champaign |
Degree: | Ph.D. |
Genre: | Dissertation |
Subject(s): | Rogers-Ramanujan functions
Rogers-Ramanujan continued fraction Ramanujan's notebooks Ramanujan's Lost Notebook theta functions modular relations partitions q-series Gollnitz-Gordon functions Gollnitz-Gordon continued fraction |
Abstract: | In his notebooks, Ramanujan recorded 40 beautiful modular relations for the Rogers-Ramanujan functions. Of these 40 identities, precisely one involves powers of the Rogers-Ramanujan functions. Ramanujan added the enigmatic note that "Each of these formulae is the simplest of a large class." This suggests that there are further modular identities involving powers of the Rogers-Ramanujan functions. Although numerous authors have studied identities for the Rogers-Ramanujan functions and various analogues, no systematic study of identities involving powers of the Rogers-Ramanujan functions has been undertaken. In this thesis, we continue the study of modular identities for the Rogers-Ramanujan functions, with particular emphasis on relations involving powers of the Rogers-Ramanujan functions. Our methods are classical, using tools and techniques that Ramanujan could have employed. These tools include, for example, manipulation of infinite series and the theory of modular equations. It is hoped that these methods will give new insights into these equations, and perhaps lead to understanding or discovering further families of identities of mathematical interest. Identities involving squares, cubes, fourth, and fifth powers of the Rogers-Ramanujan functions are enunciated and proved; many of these relations are new. Rich applications are made to the study of modular relations for the Rogers-Ramanujan continued fraction. To demonstrate the generality of our methods, analogous results are obtained in various cases for the Gollnitz-Gordon functions and the Ramanujan-Gollnitz-Gordon continued fraction. Further identities for the Rogers-Ramanujan functions, of the types found in Ramanujan's list of 40 relations for the Rogers-Ramanujan functions, are also studied. Analogous identities are obtained for the Gollnitz-Gordon functions, as well as for dodecic and sextodecic analogues of the Rogers-Ramanujan functions. |
Issue Date: | 2011-01-21 |
URI: | http://hdl.handle.net/2142/18485 |
Rights Information: | Copyright 2010 Chadwick Gugg |
Date Available in IDEALS: | 2013-01-22 |
Date Deposited: | 2010-12 |
This item appears in the following Collection(s)
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Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois -
Dissertations and Theses - Mathematics