\(q\)-difference analogue of the Euler-Poisson-Darboux equation and its Laplace sequence (Q1912843)

A \(q\)-difference analogue of the operator \[ E{(\beta, \beta')}= (x- y) \overline {E}{(\beta, \beta')}= (x-y) \partial_x \partial_y- \beta' \partial_x+ \beta \partial_y \] is considered and it is proved that \(q\)-deformation of \(E(\beta, \beta')\) is the \(q\)-difference operator: \[ E_q (\beta, \beta')= [\theta_x+ \beta ]_q [\partial_y ]_q- [\theta_y+ \beta' ]_q [\partial_x ]_q, \] where \(\overline {E} (\beta, \beta')\) is the conjugate transform of the Euler-Poisson-Darboux equation with \((x- y)^{- \beta}\).