Some important continued fractions of Ramanujan and Selberg

We provide explicit solutions for three q-difference equations which arise in Ramanujan and Selberg's work on q-continued fractions. From these solutions, we derive criteria for the convergence of three Ramanujan-Selberg continued fractions when q is a primitive m-th root of unity. Moreover, when the continued fractions converge, we determine their values explicitly. For $\vert$ q $\vert\ >$ 1, the continued fractions diverge, since the even and odd indexed convergents tend to distinct limits. We determine precisely these limits. We also give simple and uniform proofs of the three continued fraction formulas of Ramanujan and Selberg for $\vert$ q $\vert\ <$ 1.We use contiguous relations for the generalized hypergeometric series $\sb3$F$\sb2$ to give new proofs of Ramanujan's elegant continued fractions for products and quotients of gamma functions. Previous proofs were somewhat ad hoc and did not show any connections with hypergeometric functions.U of I OnlyETDs are only available to UIUC Users without author permissio