Monotonicity properties, inequalities and asymptotic expansions associated with the gamma function

We define ϑ(x) by the equality Γ(x+1)=π(xe)x(8x3+4x2+x+130)1/6eϑ(x).We call ϑ(x) the remainder of Ramanujan's formula. In this paper, we present some properties for ϑ(x), including monotonicity properties, inequalities and asymptotic expansions. Furthermore, we present some full asymptotic expansions for the gamma function related to the Nemes, Ramanujan and Burnside formulas.