CanaDAM 2021
En ligne, 25 - 28 mai 2021

Generating series and confined lattice walks - Part I
Org: Thomas Dreyfus (CNRS, Université de Strasbourg) et Andrew Elvey Price (CNRS, Université de Tours)

LUCIA DI VIZIO, CNRS, Université de Versailles-St Quentin
Differential transcendence for the Bell numbers and their relatives  [PDF]

Martin Klazar proved in 2003 that the ordinary generating function of the Bell numbers is differentially transcendental over the germs of meromorphic functions at $0$. We show that this result is an instance of a general phenomenon: on one hand we prove a general result, in a compact way, using difference Galois theory; on the other hand, we obtain as a consequence the differential transcendence of the generating functions of many other combinatorial sequences, including Bernoulli, Euler and Genocchi numbers. These results bring concrete evidence in support to the Pak-Yeliussizov conjecture. This is joint work with A.~Bostan and K.~Raschel.

HELEN JENNE, Université de Tours and Université d'Orléans
Three-dimensional lattice walks confined to an octant: non-rationality of the second critical exponent  [PDF]

We discuss the following: is there a walk model (a step set and cone $\mathcal{C}\in\mathbb{R}^d$) so the sequence $e(P,Q;n)$ of length $n$ walks in $\mathcal{C}$ from $P$ to $Q$ admits asymptotics $$e(P,Q;n)=\rho^n(K_1n^{\alpha_1}+K_2n^{\alpha_2}+\cdots)$$ with $\alpha_1\in\mathbb{Q}$ and $\alpha_2\notin\mathbb{Q}$? Indeed, there is a strong relationship between D-finiteness of the generating function $e(P,Q;n)=\sum\limits_{n\geq0}e(P,Q;n)t^n\in\mathbb{Q}[[t]]$ and the asymptotic behavior of its coefficients, and recent works study the rationality of $\alpha_1$.

In the three-dimensional case, we answer the analogous question in the continuous setting by proving there is a cone such that the heat kernel has the desired property. (Joint work with Luc Hillairet and Kilian Raschel.)

MICHAEL SINGER, North Carolina State University
Differentially Algebraic Generating Series for Walks in the Quarter Plane  [PDF]

I will present a refinement of necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, I will discuss algorithms based on the theory of Mordell-Weil lattices that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series. This is joint work with C. Hardouin and appears in arXiv:2010.00963

More Models of Walks Avoiding a Quadrant  [PDF]

We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated in [Bousquet-Mélou, 2016] and solve a new case, the king walks with all 8 nearest neighbour steps. As in the two cases solved in [Bousquet-Mélou, 2016], the associated generating function differs from a simple, explicit D-finite series (related to the enumeration of walks confined to the first quadrant) by an algebraic one. The principle of the approach is the same as in [Bousquet-Mélou, 2016], but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree. This is joint work with Mireille Bousquet-Mélou.