  In the present paper, we use difference Galois theory to study the nature of
the generating function counting walks with small steps in the quarter plane.
These series are trivariate formal power series $Q(x,y,t)$ that count the
number of walks confined in the first quadrant of the plane with a fixed set of
admissible steps, called the model of the walk. While the variables $x$ and $y$
are associated to the ending point of the path, the variable $t$ encodes its
length. In this paper, we prove that in the unweighted case, $Q(x,y,t)$
satisfies an algebraic differential relation with respect to $t$ if and only if
it satisfies an algebraic differential relation with respect $x$ (resp. $y$).
Combined with other papers, we are able to characterize the $t$-differential
transcendence of the $79$ models of walks listed by Bousquet-M\'elou and
Mishna.
