On asymptotic extrapolation

Consider a power series f∈R[[z]], which is obtained by a precise mathematical construction. For instance, f might be the solution to some differential or functional initial value problem or the diagonal of the solution to a partial differential equation. In cases when no suitable method is available beforehand for determining the asymptotics of the coefficients fn, but when many such coefficients can be computed with high accuracy, it would be useful if a plausible asymptotic expansion for fn could be guessed automatically.In this paper, we will present a general scheme for the design of such “asymptotic extrapolation algorithms”. Roughly speaking, using discrete differentiation and techniques from automatic asymptotics, we strip off the terms of the asymptotic expansion one by one. The knowledge of more terms of the asymptotic expansion will then allow us to approximate the coefficients in the expansion with high accuracy.