| In his "lost notebook" Ramanujan recorded many theorems on continued fractions. Various theorems that Ramanujan presented are discussed in this thesis.;On page 205 in his lost notebook, Ramanujan offers four explicit formulas relating two Rogers-Ramanujan continued fractions and two modular equations relating the Rogers-Ramanujan continued fraction at three arguments. We prove these identities in Chapter 2.;In Chapter 3, we prove two asymptotic formulas for two continued fractions involving the Riemann zeta function and Dirichlet L-functions which are recorded on page 45 of his lost notebook. Also, we derive asymptotic formulas for the Rogers-Ramanujan continued fraction and Gollnitz-Gordon continued fraction from our theorem.;Partial q-difference equations can be used for, among other things, generating q-continued fractions. In Chapter 4, we characterize the partial q-difference equations of arbitrary order satisfied by several families of basic hypergeometric functions.;In Chapter 5, we prove two other equations on page 45 of Ramanujan's lost notebook by using the Bauer-Muir transformation and by using the method of successive approximation. Also, we prove several continued fractions of Ramanujan which are related to the odd parts of continued fractions. |
