Section: New Results

Non-D-finite excursions in the quarter plane

The number of excursions (finite paths starting and ending at the origin) having a given number of steps and obeying various geometric constraints is a classical topic of combinatorics and probability theory. We proved in [3] that the sequence of numbers of excursions in the quarter plane corresponding to a nonsingular step set $S\subseteq {\{0,\pm 1\}}^2$ with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function of the numbers of walks with given length and prescribed ending point is not D-finite. This solves an open problem in the field of lattice path combinatorics.