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Let $W\left(u\right)\left(x\right)=1/2x^a{(1-x)}^b{u}^{\text{'}\text{'}}\left(x\right)$ with a,b ≥ 2. We consider the C₀-semigroups generated by this operator on the spaces of continuous functions, respectively square integrable functions. The connection between these semigroups together with suitable approximation processes is studied. Also, some qualitative and quantitative properties are derived.