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

TL;DR

This paper establishes a new connection between Ramanujan's asymptotic approximations and Stirling's approximation, revealing insights into the coefficients involved and using combinatorial and analytical methods.

Contribution

It uncovers a hidden link between Ramanujan's approximations and Stirling's approximation, involving Stirling numbers and Eulerian numbers, with two different proofs provided.

Findings

Established a conjectured relation between Ramanujan's approximations and the exponential integral.

Connected coefficients in Stirling's approximation to Ramanujan's approximation coefficients.

Provided a second, more analytic proof of the main relation.

Abstract

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