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References in periodicals archive ?
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
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.
The number, [E.sub.n], of walks taking steps in E and staying in the positive quarter plane grows asymptotically as [E.sub.n] = [[kappa].sub.E] x [4.sup.n] + O([(1 + 2[square root of 2]).sup.n]).
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.
Non-D-finite excursions in the quarter plane. Pre-print, http://arxiv.org/abs/1205.3300.
The 2D walks that we consider are confined to the quarter plane [N.sup.2], they join the origin of [N.sup.2] to an arbitrary point (i, j) e [N.sup.2], and are restricted to a fixed subset [??] of the step set {[??], [left arrow], [??], [up arrow], [??], [right arrow], [??], [down arrow]}.
For walks in the quarter plane, it is conjectured in [29, Section 3] that D-finiteness is preserved under reversing arrows, i.e., the generating function for a step set [??] is D-finite if and only if the generating function for the step set [??]' is, when [??]' is obtained from [??] by reversing all arrows.
Counting walks in the quarter plane. In Mathematics and computer science, II (Versailles, 2002), Trends Math., pages 49-67.