On the recurrence coefficients for the q-Laguerre weight and discrete Painlevé equations

We study the dependence of recurrence coefficients in the three-term recurrence relation for orthogonal polynomials with a certain deformation of the q-Laguerre weight on the degree parameter n. We show that this dependence is described by a discrete Painlev & eacute; equation on the family of A5(1) Sakai surfaces, but this equation is different from the standard examples of discrete Painlev & eacute; equations of this type and instead is a composition of two such. This case study is a good illustration of the effectiveness of a recently proposed geometric identification scheme for discrete Painlev & eacute; equations.