On the recurrence coefficients for the $q$-Laguerre weight and discrete Painlev\'e 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\'e equation on the family of $A_{5}^{(1)}$ Sakai surfaces, but this equation is different from the standard examples of discrete Painlev\'e 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\'e equations.