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

Generalized Ces\`aro operators, fractional finite differences and Gamma functions