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Quicksilver Solutions of a q-difference first Painlevé equation

In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a $q$-difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlevé equation ($q\Pon$), whose phase space (space of initial values) is a rational surface of type $A_7^{(1)}$. We describe four families of almost stationary behaviours, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain $q$-domain. The method, while demonstrated for $q\Pon$, is also applicable to other $q$-difference Painlevé equations.