A Generalized Fractional Calculus and Integral Transforms

In this paper a generalized fractional calculus and its applications to different topics in analysis, especially to some integral transforms, are discussed. The kernel-function of the generalized operators of integration of fractional multiorder considered here is a suitably chosen case of Meijer’s G-function: 1 $${\text{G}}_{{\text{pq}}}^{{\text{mn}}} \left[ {\sigma \left| {\begin{array}{*{20}c} {{\text{a}}_{\text{1}} , \ldots ,{\text{a}}_{\text{p}} } \\ {{\text{b}}_{\text{1}} , \ldots ,{\text{b}}_{\text{q}} } \\ \end{array} } \right.} \right] = \frac{1} {{2\pi {\text{i}}}}\int\limits_L {\frac{{\prod\limits_{{\text{k}} = {\text{1}}}^{\text{m}} {\Gamma \left( {{\text{b}}_{\text{k}} - {\text{s}}} \right)} \prod\limits_{{\text{j}} = {\text{1}}}^{\text{n}} {\Gamma \left( {{\text{1}} - {\text{a}}_{\text{j}} + {\text{s}}} \right)} }} {{\prod\limits_{{\text{k}} = {\text{m}} + {\text{1}}}^{\text{q}} {\Gamma \left( {1 - {\text{b}}_{\text{k}} + {\text{s}}} \right)} \prod\limits_{{\text{j}} = {\text{n}} + {\text{1}}}^{\text{p}} {\Gamma \left( {{\text{a}}_{\text{j}} - {\text{s}}} \right)} }}} \sigma ^{\text{s}} {\text{ds}}\quad \left( {\left[ {\text{1}} \right],\left[ 2 \right]} \right).$$