Theory of Hypergeometric Functions

Abstract

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MONODROMY AND PFAFFIAN OF LAURICELLA^|^apos;S FD IN TERMS OF THE INTERSECTION FORMS OF TWISTED (CO)HOMOLOGY GROUPS

“…We regard this correspondence as a Q(λ)-linear isomorphism from the trivial vector bundle x∈U H 1 (C • (x), ∂ u ) to Sol(ẋ). For details, refer to [AK,Y2]. 2…”

“…In this paper, we give the monodromy representation and the Pfaffian systems of F D (a, b, c) in terms of the intersection forms of twisted (co)homology groups studied in [AK,CM,Y2]. We characterize circuit transformations and the connection form of the Pfaffian system by vanishing cycles and by vanishing forms for approaches of x to components of the singular locus of the system F D (a, b, c); see Theorems 5.1 and 7.1.…”

Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral

“…In the analogous theory of integrals on complex projective spaces [6] one can choose for such a D the "reduced" one, that is, the effective divisor all of whose integral coefficients are equal to one. In our case, however, it is indispensable to choose a non-reduced divisor D = 2[ …”

“…The twisted de Rham theory developed by Aomoto [1], [3], [6] has brought a unified treatment, and a systematic way of generalization, of various hypergeometric integrals which were invented and investigated by many authors. According to Aomoto, any such integral is interpreted as a pairing of a homology class and a cohomology class on a complex projective space P n minus an effective divisor D with coefficients in a local system of rank one, which is defined by a multi-valued function on P n ramified just along D. Moreover, knowing the structures of the corresponding homology and cohomology groups enables us not only to produce systematically a system of differential equations satisfied by such integrals but also to determine the connection formulae and the monodromy representations of such integrals [2], [4], [5].…”

INTERSECTION NUMBERS AND TWISTED PERIOD RELATIONS FOR THE GENERALIZED HYPERGEOMETRIC FUNCTION <b><i>_m</i></b><b>₊</b>₁<b><i>F</i></b><b><i>_m</i></b>

“…Twisted (co)homology groups associated with the integral representation (2) For twisted homology groups, twisted cohomology groups, and the intersection forms, refer to [1,11], or [5]. We use the same notation as in [5] and [6].…”