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**Cohomological interpretation of hypergeometric series.**
*(English)*
Zbl 0808.12006

This paper explains, once again, how to get a cohomological interpretation of generalized hypergeometric series in several variables by means of “exponential modules” [for instance, see the author and F. Loeser, Jap. J. Math., New Ser. 19, 81-129 (1993; Zbl 0796.12005)] under some restrictive hypotheses. Even if yet rather computational, the new proof given here is more elementary than the older ones. It is based on a beautiful proposition which looks like a generalized division process. (Counter)-examples are given which throw light on the situation and on the particular case where restrictive hypotheses are not satisfied.

Reviewer: G.Christol (Paris)

### MSC:

12H25 | \(p\)-adic differential equations |

33C70 | Other hypergeometric functions and integrals in several variables |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

### Citations:

Zbl 0796.12005
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\textit{B. Dwork}, Rend. Semin. Mat. Univ. Padova 90, 239--263 (1993; Zbl 0808.12006)

### References:

[1] | B. Dwork , Generalized Hypergeometric Functions , Oxford University Press ( 1990 ). MR 1085482 | Zbl 0747.33001 · Zbl 0747.33001 |

[2] | B. Dwork - F. Loeser , Hypergeometric series , Jap. J. Math. , 19 ( 1993 ), to appear. MR 1231511 | Zbl 0796.12005 · Zbl 0796.12005 |

[3] | B. Dwork - F. Loeser , Hypergeometric functions and series as periods of exponential modules , Perspectives in Math. , Academic Press , to appear. MR 1307395 · Zbl 0820.33008 |

[4] | M. Kita , On hypergeometric functions in several variables, I: New integral representations of Euler type , Jap. J. Math. , 18 ( 1992 ), pp. 25 - 74 . MR 1173830 | Zbl 0767.33009 · Zbl 0767.33009 |

[5] | M. Kita , On hypergeometric functions in several variables, II: The Wronskian of the hypergeometric function of type (n + 1, m + 1) , preprint, Kanazawa University . Zbl 0799.33009 · Zbl 0799.33009 |

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