The \(q\)-Painlevé V equation and its geometrical description (Q2716916)

The \(q\)-Painlevé V equation \((q\)-\(P_{\text{V}}\) equation) is given by NEWLINE\[NEWLINEy_{n-1} y_{n+1}= (x_n-az_n) (x_n-z_n/a)/ (1-cx_n),NEWLINE\]NEWLINE NEWLINE\[NEWLINEx_nx_{n+2} =(y_{n+1}-bz_{n+1}) (y_{n+1}- z_{n+1}/b)/ (1-dy_{n+1}),NEWLINE\]NEWLINE where \(z_n=z_0q^n\) and \(a,b,c\) and \(d\) are constants. The authors study the \(q\)-\(P_{\text{V}}\) equation and its geometrical structure. Based on the bilinear formulation they obtain the equations for the multi-dimensional \(\tau\)-functions of \(P_{\text{V}}\) equation which lives in the weight lattice of the \(A_4\) affine Weyl group. The geometrical approach introduced in this paper made possible the derivation of an equation that lives in the same space but follows a different evolution path.