Analytic models and ambiguity of context-free languages
Theoretical Computer Science
Basic analytic combinatorics of directed lattice paths
Theoretical Computer Science
Walks confined in a quadrant are not always D-finite
Theoretical Computer Science - Random generation of combinatorial objects and bijective combinatorics
Classifying lattice walks restricted to the quarter plane
Journal of Combinatorial Theory Series A
Analytic Combinatorics
Two non-holonomic lattice walks in the quarter plane
Theoretical Computer Science
NIST Handbook of Mathematical Functions
NIST Handbook of Mathematical Functions
Random Walk: A Modern Introduction
Random Walk: A Modern Introduction
Random Walks and Diffusions on Graphs and Databases: An Introduction
Random Walks and Diffusions on Graphs and Databases: An Introduction
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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 prove that the sequence (e"n^S)"n"="0 of numbers of excursions in the quarter plane corresponding to a nonsingular step set S@?{0,+/-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. Moreover, we display the asymptotics of e"n^S.