We construct a displacement operator type nonlinear coherent state and examine some of its properties. In particular it is shown that this nonlinear coherent state exhibits nonclassical properties like squeezing and sub-Poissonian behaviour.

It is shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomials. Some properties of such polynomials, e.g., recurrence relation, ladder operators, differential equations, etc., have been obtained.

We evaluate Shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials. The Bialynicki–Birula–Mycielski inequality has also been tested for a number of states.