We give a parallel proof of Barnes' first lemma and of its finite analogue. In both cases we use the Mellin transform. In the classical case, the proof avoids the residue theorem. In the finite case the Gamma function is replaced by the Gaussian sum function and the beta function by the Jacobi sum function.

The Ramanujan-Bailey identity that establishes a symmetric relation between two hypergeometric series 3F2 is extended to a symmetric relation between two hypergeometric series 4F3. We also give a symmetric model that makes this symmetry evident.

We examine the convergence and analytic properties of a continued fraction of Ramanujan and its connection to the orthogonal polynomials of Meixner-Pollaczek.

The solutions of a certain class of first order linear differential equations are shown to be either completely or absolutely monotone depending on the nature of its coefficients. This is a simple theorem which is used to deduce a number of new and interesting results dealing with the complete and absolute monotonicity of functions. In particular, a partial answer is supplied to a question posed by Askey and Pollard: “When is completely monotone?”