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Generalized hypergeometric functions. (English) Zbl 0747.33001
Oxford: Clarendon Press. 188 p. (1990).
This is a monograph comprising the author’s studies on p-adic generalized hypergeometric functions. He studies the Boyarsky principle and shows the analytic dependence of the Frobenius matrix on multiplicative parameters. Using the Laplace transform, he develops a theory to encompass the four hypergeometric functions of Appell, that of Lauricella, $$_ kF_{k-1}$$ of Erdélyi et al.
The work represents a significant improvement over the author’s studies, even in the case of $$_ 2F_ 1$$. With respect to cohomology theory, it extends results of Aomoto in the sense that the above list contains the case of non-complete intersection. The book consists of 20 short chapters including a detailed description of the algebraic foundations of the theory.
Let $$a=(a_ 1,\ldots,a_ n)$$ be multiplicative parameters lying on the hyperplane $$H:a_ 1+\cdots+a_{n1}=d_ 1a_{n1+1}+\cdots+d_{n2}a_ n$$, where $$n=n_ 1+n_ 2$$. Let $$\widehat W_ a=\hat R/\sum D_{a,i}\hat R$$ be some differential module of polynomial forms. The most novel result of this book, according to the author, is a relation between the isomorphism $$X^ u:\widehat W_{a+u}\to\widehat W_ a$$ of multiplication by a monomial and the values assumed by $$\{\ell_ j(a)\}_{j=1,\ldots,m'}$$, where $$\ell_ j$$ is a suitable set of linear forms on $$H$$ which defines the cone of support of $$\hat R$$.
In spite of the title’s suggestion, this book makes no reference to the recent theory of generalized hypergeometric functions by the Gel’fand school, which is another extension of Aomoto’s theory.
Reviewer: A.Kaneko (Komaba)

##### MSC:
 33-02 Research exposition (monographs, survey articles) pertaining to special functions 33C20 Generalized hypergeometric series, $${}_pF_q$$ 32P05 Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32-XX describing the type of problem) 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 14G20 Local ground fields in algebraic geometry