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
∞
0
a
(
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
) =
∞
Ʃ
0
exp {
n
2
/
A
-2
nz
}, where
A
≫ 1.
The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation
