In this note the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized second grade fluid within an infinite circular cylinder are determined by means of the Laplace and Hankel transforms. At time t = 0 the fluid is at rest and the motion is produced by the rotation of the cylinder, around its axis, with the angular velocity t. The velocity field and the adequate shear stress are presented under integral and series forms in terms of the generalized G-functions. Furthermore, they are presented as a sum between the Newtonian solutions and the adequate non-Newtonian contributions. The corresponding solutions for the ordinary second grade fluid and Newtonian fluid are obtained as particular cases of our solutions for β = 1, respectively α = 0 and  β= 1.