Stirling's approximation and a hidden link between two of Ramanujan's approximations

A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and integral is established. The proof involves Stirling numbers, second-order Eulerian modifications of both these, Stirling's approximation gamma function. Our work provides new information about coefficients in their connection coefficients. more analytic second main result also included an appendix.