Singular walks in the quarter plane and Bernoulli numbers
Résumé
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 $t \in (0, 1/2]$. 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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