  P{\'o}lya urns are urns where at each unit of time a ball is drawn and is
replaced with some other balls according to its colour. We introduce a more
general model: The replacement rule depends on the colour of the drawn ball and
the value of the time (mod p). We discuss some intriguing properties of the
differential operators associated to the generating functions encoding the
evolution of these urns. The initial partial differential equation indeed leads
to ordinary linear differential equations and we prove that the moment
generating functions are D-finite. For a subclass, we exhibit a closed form for
the corresponding generating functions (giving the exact state of the urns at
time n). When the time goes to infinity, we show that these periodic P{\'o}lya
urns follow a rich variety of behaviours: their asymptotic fluctuations are
described by a family of distributions, the generalized Gamma distributions,
which can also be seen as powers of Gamma distributions. En passant, we
establish some enumerative links with other combinatorial objects, and we give
an application for a new result on the asymptotics of Young tableaux: This
approach allows us to prove that the law of the lower right corner in a
triangular Young tableau follows asymptotically a product of generalized Gamma
distributions.
