A hidden link between two of Ramanujan’s approximations

Abstract: In consecutive notebook entries, Ramanujan gave asymptotic approximations to the exponential function and the exponential integral. The asymptotic expansion coefficients seem to agree up to an alternating sign, as we conjectured in an earlier paper. We establish this hidden link with a combinatorial proof that involves Stirling numbers, second-order Eulerian numbers, modifications of both of these, and Stirling's approximation to the gamma function. An analytic second proof has also been provided by a referee. It is not yet clear if this result is an oddity or part of a broader picture.

combinatoricsnumber theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANY 2023)