Let
f
(
z
)
f(z)
be in
1
+
z
Q
[
[
z
]
]
1+z\mathbb {Q}[[z]]
and
S
\mathcal {S}
be an infinite set of prime numbers such that, for all
p
∈
S
p\in \mathcal {S}
, we can reduce
f
(
z
)
f(z)
modulo
p
p
. We let
f
(
z
)
∣
p
f(z)_{\mid p}
denote the reduction of
f
(
z
)
f(z)
modulo
p
p
. Generally, when
f
(
z
)
f(z)
is D-finite,
f
∣
p
(
z
)
f_{\mid p}(z)
is algebraic over
F
p
(
z
)
\mathbb {F}_p(z)
. It turns out that if
f
∣
p
(
z
)
f_{\mid p}(z)
is a solution of a polynomial of the form
X
−
A
p
(
z
)
X
p
l
X-A_p(z)X^{p^l}
, we can use this type of equations to obtain results of transcendence and algebraic independence over
Q
(
z
)
\mathbb {Q}(z)
. In the present paper, we look for conditions on the differential operators annihilating
f
(
z
)
f(z)
to guarantee the existence of these particular equations. Suppose that
f
(
z
)
f(z)
is solution of a differential operator
H
∈
Q
(
z
)
[
d
/
d
z
]
\mathcal {H}\in \mathbb {Q}(z)[d/dz]
having a strong Frobenius structure for all
p
∈
S
p\in \mathcal {S}
and we also suppose that
f
(
z
)
f(z)
annihilates a Fuchsian differential operator
D
∈
Q
(
z
)
[
d
/
d
z
]
\mathcal {D}\in \mathbb {Q}(z)[d/dz]
such that zero is a regular singular point of
D
\mathcal {D}
and the exponents of
D
\mathcal {D}
at zero are equal to zero. Our main result states that, for almost every prime
p
∈
S
p\in \mathcal {S}
,
f
∣
p
(
z
)
f_{\mid p}(z)
is solution of a polynomial of the form
X
−
A
p
(
z
)
X
p
l
X-A_p(z)X^{p^l}
, where
A
p
(
z
)
A_p(z)
is a rational function with coefficients in
F
p
\mathbb {F}_p
of height less than or equal to
C
p
2
l
Cp^{2l}
with
C
C
a positive constant that does not depend on
p
p
. We also study the algebraic independence of these power series over
Q
(
z
)
\mathbb {Q}(z)
.