Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Bostan, Alin; Chyzak, Frédéric; Hoeij, Mark van; Kauers, Manuel; Pech, Lucien

doi:10.1016/j.ejc.2016.10.010

Cited by 43 publications

(68 citation statements)

References 42 publications

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“…These methods cannot be used to determine the leading constant, or to determine terms beyond the dominant growth. In two and three dimensions there are several approaches for asymptotic enumeration of lattice models which pass through differential equations, see [3] and the references therein. Differential equation approaches become computationally infeasible in higher dimensions, and present theory does not permit treatment of dimension as a symbolic parameter.…”

Section: Reflectable Walksmentioning

confidence: 99%

The asymptotics of reflectable weighted walks in arbitrary dimension

Mishna

Simon

2020

Advances in Applied Mathematics

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Gessel and Zeilberger generalized the reflection principle to handle walks confined to Weyl chambers, under some restrictions on the allowable steps. For some models that are invariant under the Weyl group action, they express the counting function for the walks with fixed starting and endpoint as a constant term in the Taylor series expansion of a rational function. Here, we focus on the simplest case, the Weyl groups A d 1 , which correspond to walks in the first orthant N d taking steps from a subset of {±1, 0} d which is invariant under reflection across any axis. The principle novelty here is the incorporation of weights on the steps and the main result is a very general theorem giving asymptotic enumeration formulas for walks that end anywhere in the orthant. The formulas are determined by singularity analysis of multivariable rational functions, an approach that has already been successfully applied in numerous related cases.

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Section: Reflectable Walksmentioning

confidence: 99%

The asymptotics of reflectable weighted walks in arbitrary dimension

Mishna

Simon

2020

Advances in Applied Mathematics

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“…Furthermore, the kernel method implies (see, for example, Bousquet-Mélou and Mishna [13] or Bostan et al [6]) that…”

Section: Case 1: S = {N W Se}mentioning

confidence: 99%

“…Because the class of multivariate rational diagonals contains the class of algebraic functions, and is contained in the class of D-finite functions, the techniques of analytic combinatorics in several variables offer tools to investigate the connection problem (see Melczer [36] for an in-depth look at this approach). For instance, Bostan et al [6] give annihilating differential equations for each lattice path generating function in Table 2, even representing them in terms of explicit hypergeometric functions; however, they were not able to prove all asymptotics in that table, because of the connection problem. For instance, they show [6, Conjecture 2] that the number of walks with step set S = {(0, −1), (−1, 1), (1, 1)} has dominant asymptotics of the form…”

Section: Decidability Of Asymptoticsmentioning

confidence: 99%

Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior

Melczer¹

2019

SIAM J. Discrete Math.

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We consider the enumeration of walks on the non-negative lattice N d , with steps defined by a set S ⊂ {−1, 0, 1} d \{0}. Previous work in this area has established asymptotics for the number of walks in certain families of models by applying the techniques of analytic combinatorics in several variables (ACSV), where one encodes the generating function of a lattice path model as the diagonal of a multivariate rational function. Melczer and Mishna obtained asymptotics when the set of steps S is symmetric over every axis; in this setting one can always apply the methods of ACSV to a multivariate rational function whose set of singularities is a smooth manifold (the simplest case). Here we go further, providing asymptotics for models with generating functions that must be encoded by multivariate rational functions having non-smooth singular sets. In the process, our analysis connects past work to deeper structural results in the theory of analytic combinatorics in several variables. One application is a closed form for asymptotics of models defined by step sets that are symmetric over all but one axis. As a special case, we apply our results when d = 2 to give a rigorous proof of asymptotics conjectured by Bostan and Kauers; asymptotics for walks returning to boundary axes and the origin are also given.

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“…Then, F (x, y, z; t) can be written as the diagonal of a rational series, and is therefore D-finite [4,10]. The orbit sum technique gives a uniform method for proving that some octant models have D-finite generating functions.…”

Section: The Orbit Sum Methodsmentioning

confidence: 99%

“…The theory of multivariate formal Laurent series is also useful for this step: in [4], for example, this theory is used to prove computationally guessed annihilating differential operators for the sections F (x, 0; t) and F (0, y; t) of certain quadrant models. These operators lead to an annihilating differential operator for F (x, y; t), as well as explicit expressions for F (x, y; t) in terms of hypergeometric functions.…”

Section: Multivariate Formal Laurent Seriesmentioning

confidence: 99%