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Multiple Gaussian hypergeometric series. (English) Zbl 0552.33001
Ellis Horwood Series in Mathematics and Its Applications. Chichester: Ellis Horwood Limited; New York etc.: Halsted Press: a division of John Wiley & Sons. 425 p. £ 37.50 (1985).
Since the mid 1970’s, there has been a great upsurge of interest from the point of view of classical analysis in the theory of hypergeometric functions in general and special functions in particular, when it came gradually to be realised that the application of computers, although often useful, may be strictly limited without appropriate theoretical support. It is also evident that while hypergeometric functions are very frequently associated with a wide variety of symmetry groups, a classical analytic approach is still necessary when it is desired to carry out a detailed discussion of the elements of such groups which may admit of a hypergeometric representation as frequently arises in problems relating to theoretical physics.
Between the years 1953 to 1976, while a few works on the general theory of hypergeometric functions appeared, comparatively little interest in the subject was shown. However, since that date, the number of such publications has gradually increased. This area of study is extremely rich and there still remains a wide scope for further work in the field.
The book under review is devoted in the main, to the very important study of the classification and convergence conditions of multiple hypergeometric functions the dimension of which exceeds two. It consists of nine chapters as follows: - Chapter One details the background to the theory of hypergeometric series, with a brief discussion of a number of possible applications of many of the series considered later in the book. In Chapter Two, the classification of double hypergeometric series is discussed and certain of the disadvantages of the normally accepted mode of classification are pointed out. At the same time, the difficulty of presenting a generally more unified classification of hypergeometric functions of several variables is mentioned.
Chapter Three is devoted to the construction of the set of all distinct triple hypergeometric series and then to the tabulation of definitions and regions of convergence of the resulting 205 distinct series in three variables. Proofs of these regions of convergence are then made the subject of chapters four to eight, in which a general proof scheme and suitable sub-divisions of triple series into types is given. This enables the authors to treat related proofs in such a manner that the repetition of parallel reasoning is avoided.
In conclusion, in Chapter Nine a brief review of the problems of convergence, classification and notation of such series is made, and what is, in the opinion of the reviewer, particularly welcome, an indication of future avenues of development in subsequent researches into the general theory of multiple hypergeometric series is made.
This book is an invaluable addition to the literature, both from the standpoint of its lucid mathematical content, and also from the point of view of its general presentation, which are up to the highest scholastic and editorial standards. These remarks apply most particularly to the rich bibliography which includes many up-to-date entries. This work is of great potential value to all workers in the field of hypergeometric functions as related to both pure and applied mathematics.
Reviewer: H.Exton

##### MSC:
 33C05 Classical hypergeometric functions, $${}_2F_1$$ 33-02 Research exposition (monographs, survey articles) pertaining to special functions