Sharp inequalities and asymptotic expansions for the gamma function (Q897553)

The authors give a recurrence relation for determining the coefficients \(a_j\) such that \[ \Gamma(x+1)\sim\sqrt{2\pi x}(x/e)^x\Biggl(1+{1\over 12x+\sum^\infty_{j=0} a_j x^{-j}}\Biggr),\;x\to\infty. \] They also show that the best constants \(a\) and \(b\) such that \[ \sqrt{2\pi n}(n/e)^n\Biggl(1+{1\over 12n-a}\Biggr)< n!<\sqrt{2\pi n}(n/e)^n\Biggl(1+{1\over 12n-b}\Biggr) \] for all \(n\geq 1\) are \(a=0.156927\dots\) and \(b=1/2\).