  For a second-order linear differential equation with two irregular singular
points of rank three, multiple Laplace-type contour integral solutions are
considered. An explicit formula in terms of the Stokes multipliers is derived
for the characteristic exponent of the multiplicative solutions. The Stokes
multipliers are represented by converging series with terms for which limit
formulas as well as more detailed asymptotic expansions are available. Here
certain new, recursively known coefficients enter, which are closely related to
but different from the coefficients of the formal solutions at one of the
irregular singular points of the differential equation. The coefficients of the
formal solutions then appear as finite sums over subsets of the new
coefficients. As a by-product, the leading exponential terms of the asymptotic
behaviour of the late coefficients of the formal solutions are given, and this
is a concrete example of the structural results obtained by Immink in a more
general setting. The formulas displayed in this paper are not of merely
theoretical interest, but they also are complete in the sense that they could
be (and have been) implemented for computing accurate numerical values of the
characteristic exponent, although the computational load is not small and
increases with the rank of the singular point under consideration.
