  The aim of this paper is to treat the constant coefficients
functional-differential equation $y'(x)=ay(qx)+by(x)$ with the help of the
analytic theory of linear $q$-difference equations. When $ab\not=0$, the
associated Cauchy problem with $y(0)=1$ admits a unique power series solution,
which is the Hadamard product of a usual-hypergeometric series by a
basic-hypergeometric series. By means of $\theta$-modular relation, it is
proved that this entire function can be expressed as linear combination of all
the elements of a system of canonical fundamental solutions at infinity. A
family of power series related to values of Gamma function at vertical lines is
then introduced, and what really surprises us is that these explicit
non-lacunary power series possess a natural boundary. When $a\not=0$ and $b=0$,
the asymptotic behavior of solutions will be formulated in terms of the Lambert
$W$-function.
