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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)

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