# Monodromy representation associated with the Appell hypergeometric function F1 from the viewpoint of the integrals

#### Abstract

4 Classification of monodromy representations in reducible cases 16 4.1 Symmetry of the monodromy representations . . . . . . . . . . . . . . . . . . . . 16 4.2 Monodromy representation when one and only one of λi’s is an integer . . . . . . 20 4.2.1 Monodromy representaion of the case (+, ∗, ∗, ∗, ∗) . . . . . . . . . . . . . 20 4.2.2 Monodromy representaion of the case (−, ∗, ∗, ∗, ∗) . . . . . . . . . . . . . 21 4.3 Monodromy representation when two of λi’s are integers . . . . . . . . . . . . . . 23 4.3.1 Monodromy representation of the case (+,+, ∗, ∗, ∗) . . . . . . . . . . . . 23 4.3.2 Monodromy representation of the case (−,+, ∗, ∗, ∗) . . . . . . . . . . . . 24 4.3.3 Monodromy representation of the case (−,−, ∗, ∗, ∗) . . . . . . . . . . . . 24 4.4 Monodromy representation when three of λi’s are integers . . . . . . . . . . . . . 26 4.4.1 Monodromy representation of the case (−,+,+, ∗, ∗) . . . . . . . . . . . . 26 4.4.2 Monodromy representation of the case (−,−,+, ∗, ∗) . . . . . . . . . . . . 26 4.4.3 Monodromy representations for the type (+3, ∗2) . . . . . . . . . . . . . . 26 4.4.4 Monodromy representations for the type (−3, ∗2) . . . . . . . . . . . . . . 31 4.5 Monodromy representation when all of λi’s are integers . . . . . . . . . . . . . . 33 4.5.1 Monodromy representation for the type (+4,−1) . . . . . . . . . . . . . . 34 4.5.2 Monodromy representation for the type (+3,−2) . . . . . . . . . . . . . . 34 4.5.3 Monodromy representation for the type (+2,−3) . . . . . . . . . . . . . . 34 4.5.4 Monodromy representation for the type (+1,−4) . . . . . . . . . . . . . . 35 4.6 Finiteness of the reducible monodromy groups . . . . . . . . . . . . . . . . . . . . 38 4.6.1 Infiniteness of the projective monodromy groups when one and only one of the parameters λi’s is an integer . . . . . . . . . . . . . . . . . . . . . . 38 4.6.2 Infiniteness of the projective monodromy groups when two and only two of the parameters λi’s are integers . . . . . . . . . . . . . . . . . . . . . . 38 4.6.3 Infiniteness of the projective monodromy groups for the types (+2,−1, ∗2) and (+1,−2, ∗2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.6.4 The monodromy groups for the types (+3, ∗2) and (−3, ∗2) . . . . . . . . 39

### Cite this paper

@inproceedings{Mimachi2010MonodromyRA, title={Monodromy representation associated with the Appell hypergeometric function F1 from the viewpoint of the integrals}, author={Katsuhisa Mimachi and Takeshi Sasaki}, year={2010} }