Meromorphic solutions to theq-Painlevé equations around the origin

Analytic solutions ofq-P(A1) near its critical points

“…Classifying the holomorphic solutions is done using the power series method and using the q-Briot-Bouquet theorem A.1 to prove convergence. Ohyama [18,19] classified the meromorphic solutions of the discrete Painlevé equations q-P VI , q-P V and q-P III around the origin in this fashion. As the q-P(A 1 ) case is quite similar, we keep our discussion brief.…”

“…Moreover, Grammaticos et al's studies of singularity confinement and special solutions could be described as analytic information about solutions. Ohyama [18,19] studied analytic properties of q-Painlevé equations, classified all analytic solutions to the equations q-P VI , q-P V and q-P III around the origin and solved the associated linear connection problems. Mano [16], building on Ohyama's results, derived solutions to the equation q-P VI described by a broad range of asymptotics near the origin and likewise near infinity.…”

Asymptotics and Confluence for a Singular Nonlinearq-Difference-Differential Cauchy Problem

“…along the same halflines L γ p as in (5) where the Borel map Θ p ðu, zÞ remains holomorphic w.r.t z on D but suffers now (at most) exponential growth rate (91) of order k relatively to u on U p (and not in general q − exponential increase as it was the case for w p ). The second item is achieved in Subsection 4.5 (Theorem 30) where the existence of a formal power series vðt, zÞ = ∑ l≥0 h l ðzÞt l , with holomorphic coefficients h l ðzÞ on D, is established which stands for the common Gevrey asymptotic expansion of mixed order ð1/k ; ðq, 1ÞÞ on T p of the partial maps t ↦ v p ðt, zÞ, 0 ≤ p ≤ ς − 1, satisfying therefore similar estimates to (6).…”

“…Regarding the existence of local holomorphic solutions to nonlinear q − difference equations, we may refer to some recent works. Indeed, for meromorphic or holomorphic solutions around the origin for special type of nonlinear q − difference equations such as the q − Painlevé equations, we can mention [5,6]. Some category of nonlinear q − difference equations of the form…”

An Overview of Geometric Asymptotic Analysis of Continuous and Discrete Painlevé Equations

“…In contrast to the classical Painlevé equations, very few asymptotic investigations have been carried out into discrete versions of the Painlevé equations, classified by Sakai [31]. Three such studies are currently known: the additive difference first Painlevé equation [14], the q-discrete sixth Painlevé equation [20], and the q-discrete first Painlevé equation (qP I ) [23,16]. Nishioka [22] showed that qP I , which has initial value space A…”