Fractional Dunkl-Fokker-Planck Operators in Supersymmetric (SUSY) Quantum State under Foldy-Wouthuysen (FW) Transformation

In this paper, a generalized Dunkl-Fokker-Planck equation in (1 + 1) dimensions is formulated within a fractional calculus setting based on the Riemann-Liouville (RL) derivative. In this case, the construction employs reflection-deformed operators to represent parity-dependent structures together with nonlocal memory contributions characteristic of fractional evolution. Also, the associated Wigner-Dunkl supersymmetric (SUSY) framework is developed, providing a consistent algebraic structure that extends standard supersymmetry under fractional dynamics. Moreover, the system is examined for a harmonic oscillator potential combined with a centrifugal-type interaction, where the coupling between Dunkl deformation and fractional order produces modified dynamical behavior and effective transport properties. By applying an exact Foldy-Wouthuysen (FW) transformation, a nonlocal relativistic operator form of the system is obtained, yielding a closed fractional Dunkl structure governed by RL dynamics. In addition, a higher-order Foldy-Wouthuysen reduction is used to derive effective Dunkl-Fokker-Planck evolution equations from reflection-deformed relativistic quantum systems. In this context, the resulting formulation establishes a unified description of fractional diffusion, relativistic corrections, and Dunkl symmetry within an operator-based framework, with applications to generalized quantum transport and deformed statistical systems.