Some New Properties of the Gamma Function Based on Ramanujan’s Formula and Nemes’ Formula

Ramanujan presented the following approximation formula of the gamma function: \( \Gamma(x+1)\approx\sqrt{\pi}\left( \frac{x}{e}\right) ^{x} \left( 8x^{3}+4x^{2}+x+\frac{1}{30}\right) ^{1/6},\qquad x\to\infty. \) In this paper, we develop Ramanujan's approximation formula to derive a number of complete asymptotic expansions. We also establish several subadditive and superadditive properties of some functions which are related to the gamma function.