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Using an appropriate change of variable, the Schr\"odinger equation istransformed into a second-order differential equation satisfied by recentlydiscovered Jacobi type $X_m$ exceptional orthogonal polynomials. Thisfacilitates the derivation of infinite families of exactly solvable Hermitianas well as non-Hermitian trigonometric Scarf potentials and finite number ofHermitian and infinite number of non-Hermitian PT-symmetric hyperbolic Scarfpotentials. The bound state solutions of all these potentials are associatedwith the aforesaid exceptional orthogonal polynomials. These infinite familiesof potentials are shown to be extensions of the conventional trigonometric andhyperbolic Scarf potentials by the addition of some rational termscharacterized by the presence of classical Jacobi polynomials. All the membersof a particular family of these 'rationally extended polynomial-dependent'potentials have the same energy spectrum and possess translational shapeinvariant symmetry. The obtained non-Hermitian trigonometric Scarf potentialsare shown to be quasi-Hermitian in nature ensuring the reality of theassociated energy spectra.

Infinite families of (non)-Hermitian Hamiltonians associated with exceptional $X_m$ Jacobi polynomials