  Thimble regularisation is a possible solution to the sign problem, which is
evaded by formulating quantum field theories on manifolds where the imaginary
part of the action stays constant (Lefschetz thimbles). A major obstacle is due
to the fact that one in general needs to collect contributions coming from more
than one thimble. Here we explore the idea of performing Taylor expansions on
Lefschetz thimbles. We show that in some cases we can compute expansions in
regions where only the dominant thimble contributes to the result in such a way
that these (different, disjoint) regions can be bridged. This can most
effectively be done via Pad\'e approximants. In this way multi-thimble
simulations can be circumvented. The approach can be trusted provided we can
show that the analytic continuation we are performing is a legitimate one,
which thing we can indeed show. We briefly discuss two prototypal computations,
for which we obtained a very good control on the analytical structure (and
singularities) of the results. All in all, the main strategy that we adopt is
supposed to be valuable not only in the thimble approach, which thing we
finally discuss.
