Non-D-finite excursions in the quarter plane

Alin Bostan (1, 2) , Kilian Raschel (3, 4) , Bruno Salvy (5)

1 SPECFUN - Symbolic Special Functions : Fast and Certified
2 MSR - INRIA - Microsoft Research - Inria Joint Centre
3 LMPT - Laboratoire de Mathématiques et Physique Théorique
4 FDP - Fédération de recherche Denis Poisson
5 ARIC - Arithmetic and Computing

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 $e^{\mathfrak{S}}_n$.

Mots clés: Walks in the quarter plane; D-finite functions; Generating functions; Random walks in cones

Domaines: Combinatoire [math.CO]; Probabilités [math.PR]

Alin Bostan, Kilian Raschel, Bruno Salvy. Non-D-finite excursions in the quarter plane. Journal of Combinatorial Theory, Series A, 2013, 121, pp.45-63. ⟨10.1016/j.jcta.2013.09.005⟩. ⟨hal-00697386v1⟩