Singular walks in the quarter plane and Bernoulli numbers
Abstract
We consider singular (aka genus 0) walks in the quarter plane and their associated generating functions $Q(x, y, t)$, which enumerate the walks starting from the origin, of fixed endpoint (encoded by the spatial variables x and y) and of fixed length (encoded by the time variable t). We first prove that the previous series can be extended up to a universal value of t (in the sense that this holds for all singular models), namely t = 1/2 , and we provide a probabilistic interpretation of $Q(x, y, 1/2)$. As a second step, we refine earlier results in the literature and show that $Q(x, y, t)$ is indeed differentially transcendental for any t ∈ (0, 1/2]. Moreover, we prove that $Q(x, y, 1/2)$ is strongly differentially transcendental. As a last step, we show that for certain models the series expansion of $Q(x, y, 1/2)$ is directly related to Bernoulli numbers. This provides a second proof of its strong differential transcendence.
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BibTeX
@article{bostan2026singular,
title = {{Singular walks in the quarter plane and Bernoulli numbers}},
author = {Alin Bostan and Lucia Di Vizio and Kilian Raschel},
journal = {Pacific Journal of Mathematics},
year = {2026},
month = oct,
url = {https://hal.science/hal-05172463v2},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Singular walks in the quarter plane and Bernoulli numbers
AU - Alin Bostan
AU - Lucia Di Vizio
AU - Kilian Raschel
PY - 2026
DA - 2026/10/22
JO - Pacific Journal of Mathematics
UR - https://hal.science/hal-05172463v2
AB - We consider singular (aka genus 0) walks in the quarter plane and their associated generating functions $Q(x, y, t)$, which enumerate the walks starting from the origin, of fixed endpoint (encoded by the spatial variables x and y) and of fixed length (encoded by the time variable t). We first prove that the previous series can be extended up to a universal value of t (in the sense that this holds for all singular models), namely t = 1/2 , and we provide a probabilistic interpretation of $Q(x, y, 1/2)$. As a second step, we refine earlier results in the literature and show that $Q(x, y, t)$ is indeed differentially transcendental for any t ∈ (0, 1/2]. Moreover, we prove that $Q(x, y, 1/2)$ is strongly differentially transcendental. As a last step, we show that for certain models the series expansion of $Q(x, y, 1/2)$ is directly related to Bernoulli numbers. This provides a second proof of its strong differential transcendence.
SN - 0030-8730
SN - 1945-5844
ER -