特殊函数、分数阶微积分及函数论都是现代数学的重要分支，也是人们探究自然现象、研究工程问题、开展数值模拟的重要工具。本项目将主要研究由特殊函数、分数阶微积分及函数论所形成的交叉领域中的一些热点课题。我们的最终目标是试图解决著名保加利亚数学家Virginia Kiryakova教授于2018年提出的2个公开问题。这些问题关注由Prabhakar函数与Fox-Wright广义超几何函数所生成的Gel'fond-Leont'ev算子的积分表示与分解结构。它们还与几何函数论中的某些课题有着密切的联系。本项研究分为三部分。首先，我们将建立有关广义分数阶微积分的一些新的运算法则，以使我们能够得到关于多变量超几何函数的一些新公式。其次，我们将研究以单变量或多变量超几何函数为核的广义分数阶算子，并充分阐释它们与Gel’fond-Leont’ev算子之间的关系。最后，我们将转向研究Kiryakova的问题。

Special functions, fractional calculus and function theory are important branches of modern mathematics. They are also important tools for people to explore natural phenomena, study engineering problems and carry out numerical simulations. In this project, we shall mainly study some hot topics arising from the interdisciplinary field formed recently by special functions, fractional calculus and function theory. Our final goal is to try to solve two open problems raised in 2018 by the famous Bulgarian mathematician Professor Virginia Kiryakova. These problems are concerned with the integral representations and decompositional structures of the Gel'fond-Leont'ev operators generated by the Prabhakar function and the Fox-Wright generalized hypergeometric function. They are closely connected with some topics in geometric function theory. The research is divided into three parts. We shall first establish some new rules for generalized fractional calculus, which enable us to derive some new formulas of hypergeometric functions of several variables. Secondly, we shall investigate some generalized fractional operators with hypergeometric function of one or several variables as kernels. Their relations with Gel’fond-Leont’ev operators will be fully studied. Finally, we shall turn to Kiryakova’s problems.