Stokes’s phenomenon for superfactorial asymptotic series

Superfactorial series depending on a parameter are those whose terms a(n, z) grow faster than any power of n!. If the terms get smaller before they increase, the function F(z) represented by Ʃ∞0a(n, z) will exhibit a Stokes phenomenon similar to that occurring in asymptotic series whose divergence is merely factorial: across ‘Stokes lines’ in the Z plane, where the late terms all have the same phase, a small exponential switches on in the remainder when the series is truncated near its least term. The jump is smooth and described by an error function whose argument has a slightly more general form than in the factorial case. This result is obtained by a method which is heuristic but applies to superfactorial series where Borel summation fails. Several examples are given, including an analytical interpretation of the sum, and a numerical test of the error-function formula, for the function represented by F(Z) = ∞Ʃ0 exp {n2/A-2nz}, where A ≫ 1.