Apparaît dans la collection : Combinatorics and Arithmetic for Physics : Special Days

I will explain the phenomenon of resurgence in a (apparently) new ex- ample related to Stirling formula, and its generalization to quantum dilogarithm. Let us define rational Stirling numbers (St_k) = (1, 1/12, 1/288, . . . ) as coeffi- cients in the asymptotic expansion of the normalized factorial: $n ! \sim \sqrt{2\pi n} n^n e^{-n} (1 + \frac{1}{12n} + \frac{1}{288n²} - \frac{139}{51849n³} + \cdots)$ Then the asymptotic behavior of St_k for large even k is controlled by numbers St_k for small odd k, and vice versa. In the case of quantum dilogarithm one deforms Stirling numbers to Euler poly- Nomials.