A new asymptotic series for the gamma function (Q2497545)

The Stirling formula is known as \[ s! = \sqrt{2 \pi s} \left(\frac{s}{e}\right)^s e^{\gamma (s)} = \sqrt{2 \pi} \left(\frac{s}{e}\right)^s e^{\theta (s) / 12 s} \] where \(\theta (s)\) is an analytic function that satisfies \(0 < \theta(s) < 1\) for all real numbers \(s\geq 1\). In this paper, the authors replace \(\gamma (s)\) by the following convergent asymptotic series \[ \gamma (s) = \sum_{k=1}^{\infty} \frac{a_k (v)}{(s+v)(s+v+1) \ldots (s+v+k-1)} \] where \[ a_k (v) = \frac{1}{2k} \int_0^1 (1-2t)(v-t)(v+1-t) \ldots (v+k-1 - t) \, dt \] in order to get a new approach.