  In this paper, we present a complete spectral research of generalized
Ces\`aro operators on Sobolev-Lebesgue sequence spaces. The main idea is to
subordinate such operators to suitable $C_0$-semigroups on these sequence
spaces. We introduce that family of sequence spaces using the fractional finite
differences and we prove some structural properties similar to classical
Lebesgue sequence spaces. In order to show the main results about fractional
finite differences, we state equalities involving sums of quotients of Euler's
Gamma functions. Finally, we display some graphical representations of the
spectra of generalized Ces\`aro operators.
