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

Abstract

Two-by-two matrix functions, which are the lifts of the local solutions of the matrix hypergeometric differential equation of $SL$ type at $0,1,\mathrm{\infty }$ to the upper half plane by the lambda function, are introduced. Each component of these matrix functions is represented by a definite integral with a power product of theta functions as integrand, which we call in this paper Wirtinger integral. Transformations of the matrix functions under some modular transformations are established by exploiting classical formulas of theta functions. These are regarded as formulas of monodromy or connection of the hypergeometric function of Gauss.