In this paper, we construct and analyse Bessel and Flett potentials associated with the heat and Poisson semigroups in the framework of the (Formula presented.) -generalized Fourier transform. We establish fundamental properties of these potentials and derive an explicit inversion formula for the Flett potential using a wavelet-like transform. Furthermore, we introduce a β-semigroup (Formula presented.), which enables the formulation of an inversion formula for the Riesz potential. As a unifying extension, we define and investigate bi-parametric potentials (Formula presented.), which generalize both the Bessel potential and the Flett potential. In addition, we define the associated function spaces.