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Please can someone tell me more about $\delta$ -derivative ($\delta=x\dfrac{d}{dx}$) as it appears in the Hadamard definition of frational derivative or elsewhere. Why, when or where we use it. Does anything the usual derivative ($\dfrac{d}{dx}$) does that the $\delta$ -derivative does?

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I was taught its name was the $\theta$ operator, or homogeneity operator. You can easily see that its eigenfunctions are the monomials $1,x,x^2, \dots$ This operator appears in Euler's homogeneous function theorem.

You can find a little more here

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