Jacobi exceptional orthogonal polynomials for extended Scarf I potentials with position-dependent mass

We show that the Scarf I potential problem in a position-dependent mass background of the type $m(\alpha ;x)=(1+\alpha \sin x{)}^{-2}$, $0<\alpha <1$, can be solved by using a point canonical transformation mapping the corresponding Schr" odinger equation onto that of the Scarf I potential with constant mass. The inverse point canonical transformation then provides some exactly-solvable rational extensions of the Scarf I potential with positive-dependent mass associated with $X_m$-Jacobi exceptional orthogonal polynomials of type I, II, or III. The Scarf I potential problem with position-dependent mass is shown to exhibit a deformed shape invariance property in a deformed supersymmetric framework. Such a property is also valid for extended potentials of type I and II. The results are illustrated with a simple example.