On the asymptotic expansion of $\Gamma(x)$, Lagrange's inversion theorem and the Stirling coefficients

TL;DR

This paper derives an asymptotic expansion for the gamma function using Lagrange's inversion theorem and introduces a new closed-form for Stirling coefficients, enhancing understanding of gamma function approximations.

Contribution

It presents a novel derivation of the gamma function's asymptotic expansion via Lagrange's inversion theorem and offers a potentially new closed-form for Stirling coefficients.

Findings

Derived the asymptotic expansion of $\Gamma(x)$ using Lagrange's inversion theorem.

Provided a new closed-form expression for Stirling coefficients.

Connected the expansion to existing results by Boyd.

Abstract

We show how the asymptotic expansion for the gamma function $\Gamma(x)$, similar to that obtained by Boyd [Proc. Roy. Soc. London A447 (1994) 609--630], can be obtained by using a form of Lagrange's inversion theorem with a remainder. A (possibly) new closed-form representation for the Stirling coefficients is given.