open access

https://doi.org/10.14943/83823

https://hdl.handle.net/2115/69477

Transformation relations of matrix functions associated to the hypergeometric function of Gauss under modular transformations

英語

2004

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

672

1-11

In this paper we consider 2 x 2 matrix functions analytic on the upper half plane associated to the hypergeometric function of Gauss, and establish transformations of these matrix functions under some modular transformations. The matrix functions studied here are obtained as the lifts of the local solutions of the matrix hypergeometric differential equation of SL type (i.e., whose image of monodromy representation is contained in S£(2, C)) at 0, 1, oo to the upper half plane by the lambda function (§2). Each component of the matrix functions is represented by a definite integral with a power product of theta functions as integrand. Such an integral was invented by Wirtinger in order to uniformize the hypergeometric function of Gauss to the upper half plane ([5)). In this paper we call it Wirtinger integral (cf. (1.2)). In spite of many possibilities of application of the Wirtinger integral, there seems to be very few examples of application of the Wirtinger integral in literature. One of the advantages of exploiting the matrix functions above in the study of the hypergeometric function is that the monodromy property and the connection relations of the hypergeometric function are all translated as transformations of those matrix functions under modular transformations of the independent variable (§3). Moreover we can derive such transformations by exploiting classical formulas of theta functions without need to use any monodromy property or connection formula of the hypergeometric function. That is to say, this gives another new derivation of the monodromy property and the connection formulas of the hypergeometric function of Gauss.

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