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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. £39.00 (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:

${R}^{\delta }f\left(x\right)={D}^{-\delta }f\left(x\right)=\frac{1}{{\Gamma }\left(\delta \right)}{\int }_{0}^{x}{\left(x-t\right)}^{\delta -1}f\left(t\right)dt;\phantom{\rule{1.em}{0ex}}\text{Re}\left(\delta \right)>0·$

The so-called Weyl fractional integral is defined as:

${W}^{\delta }f\left(x\right)=\frac{1}{{\Gamma }\left(\delta \right)}{\int }_{x}^{\infty }{\left(t-x\right)}^{\delta -1}f\left(t\right)dt,\phantom{\rule{1.em}{0ex}}\text{Re}\left(\delta \right)>0·$

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 $\delta$ $\left({\delta }_{1}\ge 0,\cdots ,{\delta }_{m}\ge 0\right)$, $m\ge 1$, are introduced by means of kernels being ${G}_{m,m}^{m,0}$- and ${H}_{m,m}^{m,0}$-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 ${}_{p}{F}_{q}$ 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 ${H}_{m,m}^{m,0}$-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.

##### MSC:
 26A33 Fractional derivatives and integrals (real functions) 26-02 Research monographs (real functions) 33-02 Research monographs (special functions) 34B30 Special ODE (Mathieu, Hill, Bessel, etc.) 44A10 Laplace transform 30C45 Special classes of univalent and multivalent functions 45E10 Integral equations of the convolution type