Two asymptotic expansions for gamma function developed by Windschitl's formula

TL;DR

This paper derives two asymptotic expansions for the gamma function based on Windschitl's approximation, using a power series involving Bernoulli numbers, and provides more accurate related approximation formulas.

Contribution

It introduces two new asymptotic expansions for the gamma function derived from Windschitl's formula, utilizing a lesser-known power series involving Bernoulli numbers.

Findings

Derived explicit asymptotic expansions with error bounds.

Presented improved approximation formulas with higher accuracy.

Validated the expansions for all positive x.

Abstract

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} \Gamma \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1}\tfrac{\left( 2k\left( 2k-2\right) !-2^{2k-1}\right) B_{2k}}{2k\left( 2k\right) !x^{2k-1}} +R_{n}\left( x\right) \right) \end{equation*} with \begin{equation*} \left| R_{n}\left( x\right) \right| \leq \frac{\left| B_{2n}\right| }{2n\left( 2n-1\right) }\frac{1}{x^{2n-1}} \end{equation*} for all $x>0$, where $B_{2n}$ is the Bernoulli number. Moreover, we present some approximation formulas for gamma function related to Windschitl's approximation one, which have higher accuracy.