Pitman Research Notes in Mathematics Series. 301. Harlow: Longman Scientific & Technical. New York: John Wiley & Sons. x, 388 p. £ 39.00 (1994).

{\it Fractional calculus} deals with the theory of operators of integration and differentiation of arbitrary order and their applications [{\it 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); {\it S. G. Samko}, {\it A. A. Kilbas} and {\it 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 $\delta$ $(\delta_1\ge 0,\dots,\delta_m\ge 0)$, $m\ge 1$, 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.