The analytic delayed-differential equation
$z^2 \psi ^{\ \! \prime } (z) \ + \ \psi (z/q) \ = \ z$
for $q>1$
has a solution which can be expressed as a formal power series.
A $q$-advanced Laplace-Borel kernel provides for the construction
of an analytic solution whose domain is the right half plane
with vertex at the initial point $z=0$.
This method is extended to provide a continuous family of solutions,
of which a subfamily extends to a punctured neighborhood of $z=0$
on the logarithmic Riemann surface.
Conditions are given on the asymptotics of $\psi ^{\ \! \prime } (z)$ near $z=0$
to ensure uniqueness.