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Journal Articles Journal of Combinatorial Theory, Series A Year : 2014

Non-D-finite excursions in the quarter plane

Abstract

We prove that the sequence $(e^{\mathfrak{S}}_n)_{n\geq 0}$ of excursions in the quarter plane corresponding to a nonsingular step set~$\mathfrak{S} \subseteq \{0,\pm 1 \}^2$ 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. Moreover, we display the asymptotics of $e^{\mathfrak{S}}_n$.
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Dates and versions

hal-00697386 , version 1 (15-05-2012)
hal-00697386 , version 2 (05-12-2013)

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Cite

Alin Bostan, Kilian Raschel, Bruno Salvy. Non-D-finite excursions in the quarter plane. Journal of Combinatorial Theory, Series A, 2014, 121, pp.45-63. ⟨10.1016/j.jcta.2013.09.005⟩. ⟨hal-00697386v2⟩
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