  Let $f(z)\in 1 + z\mathbb{Q}[[z]]$ and $\mathcal{S}$ be an infinite set of
prime numbers such that, for all $p\in\mathcal{S}$, we can reduce $f(z)$ modulo
$p$. When $f(z)$ is D-finite, we usually get that $f(z)_{\mid p}$ is algebraic
over $\mathbb{F}_p(z)$. If in addition $f(z)_{\mid p}$ cancels a polynomial of
the form $X-A_p(z)X^{p^l}$, we can use these equations to obtain results of
transcendence and algebraic independence over $\mathbb{Q}(z)$. In this article,
we look for conditions on the differential operators canceling $f(z)$ which
guarantee the existence of these particular equations. Suppose that $f(z)$
cancels a differential operator $\mathcal{H}\in\mathbb{Q}(z)[d/dz]$ endowed
with a strong Frobenius structure for all $p\in\mathcal{S}$ as well as a
Fuchsian differential operator $\mathcal{D}\in\mathbb{Q}(z)[d/dz]$ such that
zero is a regular singular point of $\mathcal{D}$ and the zero exponents of
$\mathcal{D}$ are all zero. Our main result establishes that for almost any
$p\in\mathcal{S}$, $f(z)_{\mid p}$ cancels a polynomial of the form
$X-A_p(z)X^{p^l}$, where $A_p(z)$ is a rational fraction with coefficients in
$\mathbb{F}_p$ of height less than or equal to $Cp^{2l}$ and $C$ is a strictly
positive constant independent of $p$. We also study the algebraic independence
on $\mathbb{Q}(z)$ of these series.
