A brief story about the operators of the generalized fractional calculus. (English) Zbl 1153.26003

Kiryakova, Virginia

Fract. Calc. Appl. Anal. 11, No. 2, 203-220 (2008).

This paper is a survey of several works on “Generalized Fractional Calculus”. This notion has been first introduced by S. L. Kalla in 1969. It extends the well-known Riemann-Liouville integral \[ R^{\delta }f(z)=\frac{1}{\Gamma (\delta )}\int_{0}^{z}(z-t)^{\delta -1}f(t)dt \] to \[ If(z)=z^{-\gamma -1}\int_{0}^{z}\Phi \left( \frac{t}{z}\right) t^{\gamma }f(t)dt. \] By suitably choosing the kernel-function \(\Phi \) we can find all the existing and known fractional integrals as particular cases. In the paper, some history and some developments of this theory are briefly discussed. Also the basic results of generalized fractional calculus as well as some examples are provided.

Reviewer: Nasser-eddine Tatar (Dhahran)

MSC:

26A33	Fractional derivatives and integrals
33C60	Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)