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In the present paper, we use difference Galois theory to study the nature ofthe generating function counting walks with small steps in the quarter plane.These series are trivariate formal power series $Q(x,y,t)$ that count thenumber of walks confined in the first quadrant of the plane with a fixed set ofadmissible 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 itslength. 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 ifit satisfies an algebraic differential relation with respect $x$ (resp. $y$).Combined with other papers, we are able to characterize the $t$-differentialtranscendence of the $79$ models of walks listed by Bousquet-M\'elou andMishna.

Length derivative of the generating series of walks confined in the quarter plane