The first equation in this system characterises the generating function Q(a, u; x, y) of quarter plane walks, counted by the length (variable u), the number of NW or ES corners (variable a), and the coordinates of their endpoint (variables x and y).
Theorem 5 The generating function Q(a, u; x, y) [equivalent to] Q(x, y) of quarter plane walks is characterised by:
The generating function for quarter plane loops is thus
The equation defining Q(x, y) translates a simple recursive description of quarter plane walks, constructed by adding one or two steps at a time (see  for such arguments in the case of unweighted walks).
Tungatarov, Four boundary value problems for the Cauchy-Riemann equation in a quarter plane, Eurasian Math.
Tungatarov, Some Schwarz problems in a quarter plane, Eurasian Math.
Harutjunjan, Complex boundary value problems in a quarter plane, Complex Analysis and Applications, Proc.
n], of walks taking steps in E and staying in the positive quarter plane grows asymptotically as [E.
Theorem 10 Neither the generating function D(t) nor the generating function E(t) for walks in the quarter plane with steps from D and E, respectively, are D-finite.
Two non-holonomic lattice walks in the quarter plane.
For walks in the quarter plane, it is conjectured in [29, Section 3] that D-finiteness is preserved under reversing arrows, i.
Classifying lattice walks restricted to the quarter plane.