The asymptotic expansion for the factorial and Lagrange inversion formula

TL;DR

This paper derives explicit formulas for the coefficients in the asymptotic expansion of factorials using derivatives of elementary functions, connecting Stirling numbers of both kinds and employing Lagrange inversion.

Contribution

It provides new explicit formulas and identities for asymptotic expansion coefficients, linking Stirling numbers of the first and second kind through combinatorial analysis.

Findings

Explicit formulas for asymptotic coefficients of factorials.

New identities between Stirling numbers of both kinds.

Recurrences for the coefficients derived via Lagrange inversion.

Abstract

We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number,in terms of derivatives of powers of an elementary function. The unique explicit expression for the coefficients that appears to be known is that in the book by L. Comtet, which is given in terms of sums of associated Stirling numbers of the first kind. By considering the bivariate generating function of the associated Stirling numbers of the second kind, another expression for the coefficients in terms of them follows also from our analysis. Comparison with Comtet's expression yields combinatorial identities between associated Stirling numbers of first and second kind. It suggests by analogy another possible formula for the coefficients, in terms of a function involving the logarithm, that in fact proves to be true. The resulting coefficients, as well as the first ones are identified via the Lagrange inversion formula as the odd coefficients of the inverse of a pair of formal series, which permits us to obtain also some recurrences.