Abstract
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.
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Schedule a callMore From: Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
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