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

## TL;DR

This paper conducts a comprehensive spectral analysis of generalized Cesàro operators on Sobolev-Lebesgue sequence spaces, utilizing fractional finite differences and Gamma functions to reveal structural properties and spectra.

## Contribution

It introduces a new family of sequence spaces based on fractional finite differences and establishes their spectral properties via $C_0$-semigroups, linking to Gamma functions.

## Key findings

- Spectral properties of generalized Cesàro operators are characterized.
- Structural properties of new sequence spaces are established.
- Spectra are visualized through graphical representations.

## Abstract

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.