On a class of integral equations over C-like surfaces in \({\mathcal D}'(\Omega_+)\) algebra(s) (Q2770932)

The author defines, by analogy with the fractional differentiation through the convolution with the kernel \(x^\lambda/\Gamma(\lambda+1)\), the convolution operator \(S^M_{\lambda,\mu}f\) and claims that it is a Hilbert-Schmidt operator, continuous in the space of distributions \(\mathcal{D}'\). These operators commute and have some properties of a group: \(S^M_{\lambda,\mu}S^M_{\lambda+\mu,\alpha}=S^M_{\lambda,\mu+\alpha}\). The author calls the operator \([\Gamma(\lambda-\alpha+1)]^{-1}x^{\lambda-\alpha}S^M_{\lambda, -\alpha}\) a derivative of order \(\alpha\). Further, \(S^M_{\lambda,\mu}\) is used to define some new functionals and hypergeometric functions.