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Let 
denote the ring of
analytic 
-periodic functions
in 
on the real axis. Let
denote the ring of formal
Laurent series in 
, whose
coefficients are defined on a common strip neighbourhood of the
real axis. In this paper, we study the linear differential equation
with coefficients 
in 
. We prove that, after a change of
variables 
with 
and 
,
this equation admits a basis of 
formal solutions of the form
where 
, 
and 
.
This generalizes a well known result when 
is replaced by 
.