Tricomi problem for $q$-difference analog of anticipatory-retarding equation of mixed type

We investigate a boundary-value problem for mixed-type equation with the Lavrent'ev–Bitsadze operator in the main part and $q$-difference deviations of an argument in the lowest terms. We construct a general solution to the equation and prove a uniqueness theorem without limitations on the deviation value. The problem is solvable. We find integral representations of solutions in the elliptic and hyperbolic domains.