Differential and difference equations for recurrence coefficients of orthogonal polynomials with hypergeometric weights and Backlund transformations of the sixth Painleve equation

It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painleve VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinear difference equations. There is also a connection to the solutions of the sigma-form of the sixth Painleve equation (one of the parameters of the weights being the independent variable in the differential equation) [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painleve VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.]. In this paper, we derive a second-order nonlinear difference equation from the system and present explicit formulas showing how this difference equation arises from the Backlund transformations of the sixth Painleve equation. We also present an alternative way to derive the connection between the recurrence coefficients and the solutions of the sixth Painleve equation.