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

Abstract

Abstract. 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

Asymptotics and Confluence for a Singular Nonlinear q -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).…”

The space of monodromy data for the Jimbo-Sakai family of $q$-difference equations

“…Perhaps the most intriguing perspective is to extend our setup to q-isomonodromy problems, in particular q-difference Painlevé equations, presumably related to the deformed Virasoro algebra [35] and 5D gauge theories. Among the results pointing in this direction, let us mention a study of the connection problem for q-Painlevé VI [45] based on asymptotic factorization of the associated linear problem into two systems solved by the Heine basic hypergeometric series 2 φ 1 , and critical expansions for sollutions of q-P( A1 ) equation recently obtained in [41].…”

Elliptic asymptotics in $q$-discrete Painlevé equations

“…However, there appears to be no explicit information about such transcendental solutions, except for studies of asymptotic behaviours, which have been developed for a few cases of q-discrete Painlevé equations in the complex plane [21,28,18,11,23]. In these studies, solutions asymptotic to power series expansions in the independent variable were studied for q-P VI , q-P(A 1 ), and q-P I .…”