Geodesic
Walks in the Quarter Plane with Multiple Steps
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

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We extend the classification of nearest neighbour walks in the quarter plane to models in which multiplicities are attached to each direction in the step set. Our study leads to a small number of infinite families that completely characterize all the models whose associated group is D4, D6, or D8. These families cover all the models with multiplicites 0, 1, 2, or 3, which were experimentally found to be D-finite — with three noteworthy exceptions. 
@article{DMTCS_2015_special_285_a7,
     author = {Kauers, Manuel and Yatchak, Rika},
     title = {Walks in the {Quarter} {Plane} with {Multiple} {Steps}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
     year = {2015},
     doi = {10.46298/dmtcs.2463},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2463/}
}
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Kauers, Manuel; Yatchak, Rika. Walks in the Quarter Plane with Multiple Steps. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi : 10.46298/dmtcs.2463. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2463/

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