Modular equations and Ramanujan's cubic and quartic theories of theta functions

摘要

In this thesis, we prove several identities involving Ramanujan's general theta function. In Chapter 2, we give proofs for new Ramanujan type modular equations discovered by Somos and establish applications of some of them. In Chapter 3, we will give proofs for several Dedekind eta product identities which Somos discovered through computational searches and which Choi discovered in his work on basic bilateral hypergeometric series and mock theta functions. In Chapter 4, we derive new identities related to the Ramanujan-G"{o}llnitz-Gordon continued fraction that are similar to those for the famous Rogers-Ramanujan continued fraction. We give a new proof of the 8-dissection of the Ramanujan-G"{o}llnitz-Gordon continued fraction and also show that the signs of the coefficients of power series associated with this continued fraction are periodic with period 8. In Chapter 5, we prove several infinite series identities involving hyperbolic functions and hypergeometric functions by using the classical and quartic theories of theta functions. In Chapter 6, we study a new function called a quartic analogue of Jacobian theta functions. Finally, Chapter 7 is devoted to establishing new identities related to the Borweins' cubic theta functions and Ramanujan's general theta function. We also give equivalent combinatorial interpretations of such identities.