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Compositions of Hadamard-type fractional integration operators and the semigroup property. (English) Zbl 1027.26004
The well-known Liouville fractional integration on \(\mathbb{R}^1\) is invariant with respect to translations. The corresponding form of fractional integration on \(\mathbb{R}^1_+\) invariant with respect to dilations (obtained from the Liouville form by obvious logarithmic change of variables) is known as the Hadamard fractional integration, having been introduced by J. Hadamard in 1898. It has the form \[ J^\alpha_{0+} f=\frac{1}{\Gamma(\alpha)}\int_0^x\left(\log \frac{x}{u}\right)^{\alpha-1} \frac{f(u)du}{u}, \quad x>0 \tag{1} \] (see the presentation of this approach in the book by S. G. Samko, A. A. Kilbas and O. I. Marichev [“Fractional integrals and derivatives: theory and applications” (Russian) (1987; Zbl 0617.26004; English translation 1993; Zbl 0818.26003), Section 18.3]). The authors introduce a generalization of (1) in the form \[ J^\alpha_{0+,\mu;\gamma,\sigma}f=\frac{1}{\Gamma(\alpha)}\int_0^x\left(\frac{u}{x}\right)^\mu \left(\log \frac{x}{u}\right)^{\alpha-1} \Phi\left[\gamma,\alpha;\sigma log\frac{u}{x}\right]\frac{f(u)du}{u} , \quad x>0, \tag{2} \] keeping the kernel homogeneous of order \(-1\) so that the Mellin transform approach is applicable. The authors prove some analogues of the semigroup property for the operator (2). The right hand-sided version of (2) is also considered.

MSC:
26A33 Fractional derivatives and integrals
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