The question to have an umbral calculus
in non zero characteristic $p$, has been considered on some special subspaces of the space of polynomials,
for instance subspaces of the space of additive polynomials
with coefficients in a complete valued field of characteristic $p$ and their closure in spaces of continuous
functions.
In her PhD thesis, M. Héraoua has given an umbral calculus on the so called ring of formal differential operators
which has a coalgebra structure. In many respects, this umbral calculus is as in the classical umbral calculus in
characteristic zero.
It turns out that the technique used by M. Héraoua can be extended in the topological case. More precisely, let
$\mathbb{F}_q$ be the finite field with $q$ elements and $\mathbb{F}_q[[T]]$ be
the ring of formal power series with coefficients in $\mathbb{F}_q$. Then with the addition of formal power series
and the $T$-adic topology, $\mathbb{F}_q[[T]]$ is a totally discontinuous compact group.
Let $K$ be a complete valued field, extension of the valued field of formal Laurent series $\mathbb{F}_q((T))$, then it
is well known that the space of continuous functions $\mathcal{C}(\mathbb{F}_q[[T]] , K) $ is an ultrametric Hopf
algebra. The coalgebra structure of
$\mathcal{C}(\mathbb{F}_q[[T]] , K) $ is that of a binomial divided power
coalgebra. In a previous work we have described the algebra of difference operators of the Banach coalgebra
$\mathcal{C}(\mathbb{F}_q[[T]] , K) $. Here, we show that the Keigher-Pritchard's divided powers of an element of
the maximal ideal of the dual algebra of $\mathcal{C}(\mathbb{F}_q[[T]] , K) $ can be performed. With this, and the
fact that the algebra of difference operators is isomorphic to the latter dual algebra,
one recovers much part of classical umbral calculus.