Introduction

In this chapter, we introduce the histories of the fractional calculus fractional calculus of constant and variable orders, fractional calculus of constant and variable orders with respect to another function, general fractional calculus of constant and variable orders and general fractional calculus of constant and variable orders with respect to another function. Meanwhile, the special functions special functions contain the Gamma function Gamma function , Gauss hypergeometric function Gauss hypergeometric function , Bessel function Bessel function , Kummer function Kummer function , Kohlrausch-Williams-Watts function Kohlrausch-Williams-Watts function , Bessel-Clifford function Bessel-Clifford function , Macdonald function Macdonald function , Mittag-Leffler function Mittag-Leffler function , Wiman function, Wright function Wiman function, Wright function , Meijer G function Meijer G function , MacRobert E function MacRobert E function , Rabotnov function Rabotnov function , Agarwal function Agarwal function , Erdélyi-Magnus-Oberhettinger-Tricomi function Erdélyi-Magnus-Oberhettinger-Tricomi function , Fox H function Fox H function , Prabhakar function Prabhakar function , Miller-Ross function Miller-Ross function , Gorenflo-Mainardi function Gorenflo-Mainardi function , two-parametric Lorenzo-Hartley function two-parametric Lorenzo-Hartley function , and so on. The special functions with respect to another function are proposed for the first time.