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On critical expansions of solutions of the discrete Painleve equation $q$-$P(A_1)$ and corresponding monodromy Roffelsen, Pieter


In a recent paper [1], we classified the critical behaviour of solutions of the discrete Painleve equation $q-P(A_1)$ around the origin and infinity. Even though this equation is ranked higher than q-PVI and hence PVI in Sakai's classification scheme [2], the critical behaviours seem to resemble those found for the sixth Painleve equation. The generic expansions near the two critical points each contain two arbitrary q-constants, and relating them explicitly constitutes the $q-P(A_1)$ connection problem. A well known method of attack is given by the isomonodromic deformation method, which we apply to an associated linear system formulated by Yamada [3]. It turns out that the connection problem of the Yamada system factorises asymptotically, both as the PainlevВґe variable approaches the origin and infinity, into two copies of a linear system associated with the continuous dual q-Hahn polynomials. The connection matrix of the limiting system can be calculated explicitly, which allows us to solve the direct monodromy problem for the critical expansions of solutions of $q-P(A_1)$. In particular this reduces the $q-P(A_1)$ connection problem to solving an equation involving q-elliptic functions. References [1] N. Joshi and P. Roffelsen, Analytic solutions of $q-P(A_1)$ near its critical points, arXiv:1510.07433 [nlin.SI]. [2] H. Sakai, Rational Surfaces Associated with Affine Root Systems and Geometry of the Painleve Equations, Commun. Math. Phys. 220 (2001). [3] Y. Yamada, Lax Formalism for q-Painleve Equations with Affine Weyl Group Symmetry of Type $E_n^{(1)}$ , Int. Math. Res. Not. IMRN 2011 (2011).

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