A fractional power interpretation of the Laguerre derivative $(DxD)^\alpha,\ D\equiv {d\over dx} $ is discussed. The corresponding fractional integrals are introduced. Mapping and semigroup properties, integral representations and Mellin transform analysis are presented. A relationship with the Riemann-Liouville fractional integrals is demonstrated. Finally, a second kind integral equation of the Volterra-type, involving the Laguerre fractional integral is solved in terms of the double hypergeometric type series as the resolvent kernel.

中文翻译：

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关于 Laguerre 分数阶积分微分

讨论了 Laguerre 导数 $(DxD)^\alpha,\D\equiv {d\over dx} $ 的分数幂解释。介绍了相应的分数积分。介绍了映射和半群性质、积分表示和梅林变换分析。证明了与 Riemann-Liouville 分数积分的关系。最后，以双超几何型级数为求解核，求解了涉及拉盖尔分数阶积分的第二类Volterra型积分方程。