Exact Analytical Solutions of the Dunkl-Schr\"odinger Equation for the Deng-Fan Potential

We present exact analytical solutions for the radial Dunkl-Schr\"odinger equation (DSE) confined by the Deng-Fan molecular potential. By employing the Pekeris approximation to resolve the centrifugal singularity and applying the parametric Nikiforov-Uvarov method, we derive closed-form expressions for the energy eigenspectrum and the corresponding radial wavefunctions expressed in terms of Jacobi polynomials. Our investigation reveals that the Dunkl reflection parameter $\mu$ fundamentally alters the system's topology by breaking spatial symmetry and introducing a parity-dependent repulsive force. Numerical analysis demonstrates a monotonic increase in energy eigenvalues with increasing $\mu$, confirming an effective "hard core" behavior at the origin. The results are shown to be consistent with standard quantum mechanics in the limit $\mu \to 0$. This study establishes the Dunkl formalism as a robust tool for modeling quantum systems characterized by parity-dependent exclusion effects and strong short-range correlations.