Unified treatment of several asymptotic formulas for the gamma function (Q372854)

The author proves the asymptotic formula \[ \Gamma(x+1)\sim \sqrt{2\pi x}(x/e)^x\Biggl(1+ \sum^\infty_{j=1} b_j/x^j\Biggr)^{1/r}, \] where \(r\) is a given nonzero real number, and the coefficients \(b_j= b_j(r)\) are exactly determined in terms of finite sums involving Bernoulli numbers. This result gives a unified treatment of results due to Laplace, Ramanujan, Karatsuba, Gosper, etc.