On a \(q\)-deformation of the discrete Painlevé I equation and \(q\)- orthogonal polynomials (Q1320092)

The author obtains a \(q\)-analogue of the discrete Painlevé I equation by \(q\)-deforming the continuous isomonodromy problem for discrete PI equation. This equation has a form \[ q^{-n}\left[\gamma=\beta q^{- 2}(R_{n+1}+R_ n+R_{n-1})+\beta(q^{-2}-1)\sum^{n-2}_{j=- \infty}R_ j\right]= {(n)_ q+\alpha q^ n\over R_ n}, \tag{*} \] where \((n)_ q=(q^ n-1)/(q-1)\). Introducing the basic \(q\)-Hermite polynomials, which are orthogonal with respect to the Jackson integral, he looks for an explicit solution to the equation \((*)\) in terms of such polynomials. If \(q\to 1\) he obtains immediately the classical version of the discrete Painlevé equation.