Special solutions of a discrete Painlevé equation for quantum minimal surfaces

We consider solutions of a discrete Painlevé equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe, and Kontsevich, and in earlier work of Cornalba and Taylor on static membranes. While the discrete equation admits a continuum limit to the Painlevé I differential equation, we find that it has the same space of initial values as the Painlevé V equation with certain specific parameter values. We further explicitly show how each iteration of this discrete Painlevé I equation corresponds to a certain composition of Bäcklund transformations for Painlevé V, as was first remarked in a work by Tokihiro, Grammaticos, and Ramani. In addition, we show that some explicit special function solutions of Painlevé V, written in terms of modified Bessel functions, yield the unique positive solution of the initial value problem required for quantum minimal surfaces.