Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

In this paper, we present a new method of deducing infinite sequences of exact solutions of q-discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q-discrete version of the second Painlevé equation (q-PII) with affine Weyl group symmetry of type (A2+A1)(1). We show, for the first time, how to use the q-discrete linear problem associated with q-PII to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q-PII here, is also applicable to other discrete Painlevé equations.

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