Asymptotic expansions for the gamma function

Mortici (2015) [31] proposed a new formula for approximating the gamma function and the convergence of the corresponding asymptotic series is very fast in comparison with other classical or recently discovered asymptotic series. In this paper, by the Lagrange–Bürmann formula we give an explicit formula for determining the coefficients akak(k=1,2,…)(k=1,2,…) in Mortici's formula such thatΓ(x+1)2πx(xe)x∼exp⁡{∑k=1∞ak(x12x2+25)k},x→∞. Moreover, by the cycle indicator polynomial of symmetric group, we give an explicit expression for the coefficients bkbk(k=0,1,…)(k=0,1,…) of the following expansion:Γ(x+1)2πx(xe)x∼(∑k=0∞bk(x12x2+25)k)1/r,x→∞. A recursive formula for calculating the coefficients bkbk(k=0,1,…)(k=0,1,…) is also given.