  The classical Pieri formula gives a multiplicity free expansion of an
irreducible module with a fundamental one for the complex general linear group.
In this article we replace the tensor product by the fusion product and prove
an analogue Pieri formula for higher level Demazure modules for the affine Lie
algebra $\widehat{\mathfrak{sl}}_{n+1}$. To be more precise, we show that the
fusion product of an arbitrary stable Demazure module with a fundamental module
admits a multiplicity free excellent filtration and the successive quotients
are described explicitly. As a consequence, we derive recurrence relations for
the generating series encoding the numerical multiplicities in Demazure flags
of level one Demazure modules.
