##
**Generalized fractional calculus and applications.**
*(English)*
Zbl 0882.26003

Pitman Research Notes in Mathematics Series. 301. Harlow: Longman Scientific & Technical. New York: John Wiley & Sons. x, 388 p. (1994).

Fractional calculus deals with the theory of operators of integration and differentiation of arbitrary order and their applications [K. Nishimoto: “Fractional calculus”, Vol. I (1984; Zbl 0605.26006), Vol. II (1987; Zbl 0702.26011), Vol. III (1989; Zbl 0798.26005), and Vol. IV (1991; Zbl 0798.26006); S. G. Samko, A. A. Kilbas and O. I. Marichev: “Integrals and derivatives of fractional order and some of their applications” (Russian: 1987; Zbl 0617.26004; English translation: 1993; Zbl 0818.26003)]. The concept of differintegral of complex order \(\delta\), which is a generalization of the ordinary \(n\)th derivative and \(n\)-times integral, can be introduced in several ways. One of the simple definition of an integral of an arbitrary order is based on an integral transform, called the Riemann-Liouville operator of fractional integration:
\[
R^\delta f(x)= D^{-\delta}f(x)= {1\over\Gamma(\delta)} \int^x_0 (x-t)^{\delta- 1}f(t)dt;\quad\text{Re}(\delta)>0.
\]
The so-called Weyl fractional integral is defined as:
\[
W^\delta f(x)= {1\over\Gamma(\delta)} \int^\infty_x (t-x)^{\delta- 1}f(t)dt,\quad\text{Re}(\delta)>0.
\]
There are several modifications and generalizations of these operators, but the most widely used in applications are the Erdélyi-Kober operators.

This book is devoted to a systematic and unified development of a new generalized fractional calculus. Generalized operators of integration and differentiation of arbitrary multiorder \(\delta\) \((\delta_1\geq 0,\dots,\delta_m\geq 0)\), \(m\geq 1\), are introduced by means of kernels being \(G^{m,0}_{m,m}\)- and \(H^{m,0}_{m,m}\)-functions. Due to this special choice of Meijer’s G-function (and Fox’s H-function) in the single integral representations of the operators considered here, a decomposition into commuting Erdélyi-Kober fractional operators holds under suitable conditions. The author has developed a full chain of operational rules, mapping properties and convolutional structure of the generalized (m-tuple) fractional integrals and the corresponding derivatives.

Historical background and the theme of the book is contained in the Introduction. Chapters 1 and 2 treat the basic concepts and properties of the Erdélyi-Kober fractional integrals. Chapter 3 is devoted to the class of so-called hyper-Bessel integral and differential properties, Poisson-Sonine-Dimovski transmutations and Obrechkoff transform. Some new integral and differintegral formulas for the generalized hypergeometric functions \({_pF_q}\) are considered in Chapter 4. Some other applications of the generalized fractional calculus: Abel’s integral equation, theory of univalent functions and generalized Laplace type transforms are treated in the Chapter 5. Fractional integration operators involving Fox’s \(H^{m,0}_{m,m}\)-function are studied here in different functional spaces. To make the book self-contained, the author has given an Appendix dealing with definition and main properties of the Meijer’s G-function, Fox’s H-function, Hyper Bessel, D- and n-Bessel functions, etc. The references include 519 titles and a Citation Index is provided, showing the articles referred to in the Sections.

This book is an exposition of a self-contained new theory of generalized operators of differintegrals. This monograph is very useful for graduate students, lecturers and researchers in Applied Mathematical Analysis and related Mathematical Sciences. This book is a good addition to the existing literature on the subject, and it will stimulate more research in this new exciting field of fractional calculus.

This book is devoted to a systematic and unified development of a new generalized fractional calculus. Generalized operators of integration and differentiation of arbitrary multiorder \(\delta\) \((\delta_1\geq 0,\dots,\delta_m\geq 0)\), \(m\geq 1\), are introduced by means of kernels being \(G^{m,0}_{m,m}\)- and \(H^{m,0}_{m,m}\)-functions. Due to this special choice of Meijer’s G-function (and Fox’s H-function) in the single integral representations of the operators considered here, a decomposition into commuting Erdélyi-Kober fractional operators holds under suitable conditions. The author has developed a full chain of operational rules, mapping properties and convolutional structure of the generalized (m-tuple) fractional integrals and the corresponding derivatives.

Historical background and the theme of the book is contained in the Introduction. Chapters 1 and 2 treat the basic concepts and properties of the Erdélyi-Kober fractional integrals. Chapter 3 is devoted to the class of so-called hyper-Bessel integral and differential properties, Poisson-Sonine-Dimovski transmutations and Obrechkoff transform. Some new integral and differintegral formulas for the generalized hypergeometric functions \({_pF_q}\) are considered in Chapter 4. Some other applications of the generalized fractional calculus: Abel’s integral equation, theory of univalent functions and generalized Laplace type transforms are treated in the Chapter 5. Fractional integration operators involving Fox’s \(H^{m,0}_{m,m}\)-function are studied here in different functional spaces. To make the book self-contained, the author has given an Appendix dealing with definition and main properties of the Meijer’s G-function, Fox’s H-function, Hyper Bessel, D- and n-Bessel functions, etc. The references include 519 titles and a Citation Index is provided, showing the articles referred to in the Sections.

This book is an exposition of a self-contained new theory of generalized operators of differintegrals. This monograph is very useful for graduate students, lecturers and researchers in Applied Mathematical Analysis and related Mathematical Sciences. This book is a good addition to the existing literature on the subject, and it will stimulate more research in this new exciting field of fractional calculus.

Reviewer: S.L.Kalla (Kuwait)

### MSC:

26A33 | Fractional derivatives and integrals |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

33-02 | Research exposition (monographs, survey articles) pertaining to special functions |

34B30 | Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) |

44A10 | Laplace transform |

30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |