## Generalized hypergeometric functionsHypergeometric functions have occupied a significant position in mathematics for over two centuries. This monograph, by one of the foremost experts, is concerned with the Boyarksy principle which expresses the analytic properties of a certain proto-gamma function. The author develops a theory which is broad enough to encompass several of the most important hypergeometric functions in the literature and their cohomology. A central theme is the development of the Laplace transform in this context and its application to spaces of functions associated with hypergeometric functions. Consequently, this book represents a significant further development of the theory and demonstrates how the Boyarksy principle may be given a cohomological interpretation. The author includes an exposition of the relationship between this theory and Gauss sums and generalized Jacobi sums, and explores the theory of duality which throws new light on the theory of exponential sums and confluent hypergeometric functions. |

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### Contents

INTRODUCTION | 1 |

MULTIPLICATION BY X GAUSS CONTIGUITY | 13 |

ALGEBRAIC THEORY | 16 |

Copyright | |

21 other sections not shown

### Common terms and phrases

a e H a$Hg algebraic algebraically independent analytic annihilator assertion assume automorphism Chapter choose coefficients commutative diagram completes the proof conclude Corollary deduce define definition denote dimension dual Dwork endomorphism equation exists factors finite follows gamma function gauss gauss sums given hence hypergeometric functions implies independent indicial polynomial integral integral domain irreducible isomorphism Jacobi sums K[Xl Ker(a kernel Laplace transform Lemma lies linear forms matrix modulo monomials multiplication natural map non-trivial notation open subset operator p-adic partitions Pastro polar locus proof of Lemma Proof of Theorem Proposition R-sequence rational function relative replaced represents a basis resp right side ring satisfies 2.6 satisfying the conditions semigroup shows space specialization subring subspace surjective trivial u e H0 ueH0 v e H0 variables verify write yg(a Zariski open