Abstract. Let \(00, \] whenever the integral exists. If \(\mu=p\in\mathbb{N}\), we set \[ {}_0^{\mathrm{C}}D_x^p f := f^{(p)}. \] Lemma 2.2. Let \(\mu>0\) and \(\beta>-1\). Then \[ {}_0 I_x^\mu\!\left(\frac{x^\beta}{\Gamma(\beta+1)}\right) = \frac{x^{\beta+\mu}}{\Gamma(\beta+\mu+1)}. \] Proof. By Definition 2.1, \[ {}_0 I_x^\mu\!\left(\frac{x^\beta}{\Gamma(\beta+1)}\right) = \frac{1}{\Gamma(\mu)\Gamma(\beta+1)} \int_0^x (x-t)^{\mu-1}t^\beta\,dt. \] Substituting \(t=xu\), we obtain \[ \frac{x^{\beta+\mu}}{\Gamma(\mu)\Gamma(\beta+1)} …
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