would admit exact energy eigenvalues given by Now, let M ( ρ, ϕ, z ) = M ( ρ ) = bρ $^{2}$/ 2 move under the influence of a trigono- metric Rosen-Morse potential ˜ V ( z ) = U$_{◦}$ cot $^{2}$( πz/d ) ; z ∈ [0 , d ]. Where U$_{◦}$ and d are two positive parameters. In this case, (see Ma et al [32] for more details, notice that one should consider 2 µ = ℏ = 1 of Ma as proper parametric mapping into our settings) and Now consider the PDM-particle to have a radial Coulombic-type mass of the form M ( ρ, ϕ, z ) = M ( ρ ) = bρ − $^{1}$/ 2, (i.e., υ = − 1) and subjected to move in a radial Coulombic potential ˜ V ( ρ ) = − 2 ˜ A/ρ . In this case, E = a 2 4 b − 1 4 b   ( √$_{D/ϵ}$ − ˜ n$_{z}$ − 1 2 ) 2 n$_{ρ}$ + √ m 2 + 3) − 2 ( ζ − β $_{) + 1}$ $^{2}$ . (27) V ( ρ, ϕ, z ) = a 2 4 b + U$_{◦}$ bρ 2 cot $^{2}$( πz/d ) ; z ∈ [0 , d ] , (28) k 2 z = 1 d 2 [ Cd + ˜ n$_{z}$π ] 2 − U$_{◦}$ ; C = π 2 d ( 1 + √ 1 + 4 U$_{◦}$d 2 π 2 ) , (29) E = a 2 4 b − 1 4 b [ [ Cd + ˜ n$_{z}$π ] 2 /d 2 − U$_{◦}$ 2 n$_{ρ}$ + √ ( m 2 + 3) − 2 ( ζ − β ) + 1 ] 2 . (30) 3.2 The Radial Coulombic g ( ρ ) = bρ − $^{1}$/ 2 and equation (14) yields V ( ρ, ϕ, z ) = − ˜ A b + ρ b ˜ V ( z ) , (31)    − ∂ 2 ρ + ˜ ℓ 2 − 1 − 1 / 4 ρ 2 − 2 ( ˜ bE + ˜ A ) ρ $^{2}$   U ( ρ ) = − k 2 $_{z}$U ( ρ ) ; ˜ b = b/ 2 , (32) 11