Schlesinger transformations for linearizable equations

We introduce the Schlesinger transformations of the Gambler equation. The l atter can be written, in both the continuous and discrete cases, as a syste m of two coupled Riccati equations in cascade involving an integer paramete r n. In the continuous case, the parameter appears explicitly in the equati on, while in the discrete case it corresponds to the number of steps for si ngularity confinement. Two Schlesinger transformations are obtained relatin g the solutions for some value n to that corresponding to either n + 1 or n + 2.