Some Problems Connected with Ramanujan's Hypergeometric Series

Starting from Gauss hypergeometric function, which Ramanujan studied in detail, limiting forms are considered and pathways are constructed to go from a Gauss hypergeometric series to confluent hypergeometric series to binomial series to Bessel series and finally to exponential series. The path from binomial to exponential is discussed in detail. It is shown that this path is connected to reaction-rate probability integrals, non-extensive statistical mechanics, Tsallis statistics, superstatistics, Kratzel integrals, inverse Gaussian density Bayesian procedures and generalizations of these. Products of binomials forms are shown to be connected to Mellin convolutions, densities of products and ratios and finally to fractional integrals. Some aspects of the corresponding theory of functions of matrix argument are also discussed.