Modular identities for the Rogers-Ramanujan functions and analogues

Rogers-Ramanujan functions;Rogers-Ramanujan continued fraction;"Ramanujans notebooks";"Ramanujans Lost Notebook";theta functions;modular relations;partitions;q-series;Gollnitz-Gordon functions;Gollnitz-Gordon continued fraction

Gugg, Chadwick

关键词: Rogers-Ramanujan functions;    Rogers-Ramanujan continued fraction;    "Ramanujans notebooks";    "Ramanujans Lost Notebook";    theta functions;    modular relations;    partitions;    q-series;    Gollnitz-Gordon functions;    Gollnitz-Gordon continued fraction;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/18485/Gugg_Chadwick.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 "Eachof these formulae is the simplest of a large class." This suggests that there arefurther 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. Richapplications 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 obtainedfor the Gollnitz-Gordon functions, as well as for dodecic and sextodecic analogues of the Rogers-Ramanujan functions.