Analytic solutions of $q$-$P(A_1)$ near its critical points

For transcendental functions that solve non-linear $q$-difference equations, the best descriptions available are the ones obtained by expansion near critical points at the origin and infinity. We describe such solutions of a $q$-discrete Painlevé equation, with 7 parameters whose initial value space is a rational surface of type $A_1^{(1)}$. The resultant expansions are shown to approach series expansions of the classical sixth Painlevé equation in the continuum limit.

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