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H-transforms. Theory and applications. (English) Zbl 1056.44001
Analytical Methods and Special Functions 9. Boca Raton, FL: Chapman & Hall/CRC (ISBN 0-415-29916-0/hbk). xii, 389 p. (2004).
The book deals with $$H$$-transforms introduced by the reviewer and P. K. Mittal in [J. Aust. Math. Soc. 11, 142–148 (1970; Zbl 0193.08401)]. The importance of the present study lies in the fact that the $$H$$-function (also known as Fox $$H$$-function) that occurs as the kernel of the transform, generalizes most of the known special functions with the result that a large number of integral transforms introduced from time to time by several authors follow as simple special cases from the $$H$$-transform.
The book contains a distinctive historical survey and bibliographical remarks at the end of each chapter. This enhances the utility of the book and makes it complete but at the same time pushes the original findings of the concerned authors to the back seat.
The book consists of eight chapters. The first two contain several results from the theory of the $$H$$-function. The majority of these results are known and are available in the literature. The authors have put most of the huge amount of scattered results concerning the $$H$$-function at one place in a systematic manner. In the process, they provide alternate proofs, extend or correct the conditions of validity and point out errors in some of these recorded results. Chapters 3–5 are devoted to the study of $$L_{v,r}$$-theory of integral transforms with $$H$$-function kernels. The integral transforms with the Meijer $$G$$-function as kernel and modified $$G$$-transforms are presented in chapter 6. Laplace and Stieltjes transforms, the transforms whose kernels are Whittaker and parabolic cylinder functions, hypergeometric functions and Wright functions form the subject matter of study of Chapter 7. Bessel type integral transforms are considered in Chapter 8.
The book has a bibliography comprising nearly 600 entires. Lot of time and energy must have been spent in this work. Even now some important papers having a direct bearing on the topic of study of the book have escaped notice of the authors. Thus, out of seven special functions of practical utility that cannot be obtained from the Meijer $$G$$-function but follow as special cases of the $$H$$-function [see C. F. Lorengo and T. T. Hartley, NASA/TP-209424 (1999); K. C. Gupta, Ganita Sandesh 15(2), 63–66 (2001)] only one viz. the Mittag-Leffler function finds place in the book. A very large number of theorems exhibiting relationships between images and originals or revealing interconnections existing between images of the related functions in the integral transforms whose kernels are special cases of the $$H$$-function have been established by several authors. A reference to some of these theorems is also given in the book at the end of chapters 7 and 8 in the bibliographical remarks. A paper by the reviewer [Int. J. Math. Math. Sci. 5, 357–363 (1982; Zbl 0489.44001)] that unifies 24 of such theorems has failed to attract the attention of the authors. Unified theorems of this type involving the $$H$$-function as kernels are also recorded on pages 46–47 of the monograph by H. M. Srivastava, K. C. Gupta and S. P. Goyal [The $$H$$-functions of one and two variables with applications (1982; Zbl 0506.33007)], referred extensively in the book.
The most important application of integral transforms is in the solution of differential or integral equations subject to certain boundary conditions occurring in various fields of sciences and engineering. This aspect of applications is not covered in the book. Also, the results given in chapters 6–8 are modified, altered versions or particular cases of the corresponding results established in chapters 3–5 and cannot be considered as applications of $$H$$-transforms.
The entire subject matter contained in the book is well presented with clear and precise conditions of validity. It is mathematically sound and of good quality. The book will prove to be immensely useful to students, teachers and research workers interested in the study and development of the theory of $$H$$-functions and associated integral transforms.

##### MSC:
 44-02 Research exposition (monographs, survey articles) pertaining to integral transforms 44A20 Integral transforms of special functions 44A10 Laplace transform 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions) 33C20 Generalized hypergeometric series, $${}_pF_q$$ 26A33 Fractional derivatives and integrals 44A15 Special integral transforms (Legendre, Hilbert, etc.) 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$