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Hypergeometric functions – past, present, and possible future. (Japanese) Zbl 0787.33001
This paper written in Japanese gives a survey of recent progress on hypergeometric functions as well as a bit of the past back to Gauss and the related topics. The main subject is the connection and the integral associated with the forms $$\Phi dx_ 1 \wedge \cdots \wedge x_ n$$ and $$\exp(f_ 0)\Phi$$, where $$\Phi=f_ 1^{\lambda_ 1} \cdots f_ m^{\lambda_ m}$$; $$f_ i$$ are polynomials in the variables $$x_ 1,\dots,x_ n$$ and $$\lambda_ i$$ are complex numbers. This article shall be translated in English and be printed in the journal Sugaku. Let me here cite the title of each section. §1 The hypergeometric function of Gauss-Euler. §2 The twisted de Rham cohomology. §3 Several complexes. §4 Projective quasi-invariance. §5 Gauss-Manin connection and holonomic difference equations. §6 Contiguity relations and holonomic difference systems. §7 Irregular singularity. §8 Related topics.
Reviewer: T.Sasaki (Kobe)

##### MSC:
 33-02 Research exposition (monographs, survey articles) pertaining to special functions 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions) 33C70 Other hypergeometric functions and integrals in several variables 33C90 Applications of hypergeometric functions
##### Keywords:
Rham cohomology; holonomic difference equations