In this paper we study boundary eigenvalue problems for first order systems of ordinary differential equations of the form \[zy'\left( z \right) = \left( {\lambda A_1 \left( z \right) + A_0 \left( z \right)} \right)y\left( z \right),\,\,y\left( {ze^{2\pi i} } \right) = e^{2\pi iv} y\left( z \right)\] for z ϵ Slog, where S is a ring region around zero, Slog denotes the Riemann surface of the logarithm over S, the coefficient matrix functions A1(z) and A0(z) are holomorphic on S, and v is a complex number. The eigenfunctions of this eigenvalue problem are the Floquet solutions of the differential system with v as characteristic exponent. For an open subset S0 of S, the notion of A1-convexity of the pair (S0, S) is introduced. For A1-convex pairs (S0, S) it is shown that the expansion into eigenfunctions and associated functions of holomorphic functions on Slog, satisfying the monodromy condition y(ze2πi) = e2πivy(z), converges regularly on Slog0 and is unique. If S is a pointed neighbourhood of 0 and A1(z) is holomorphic in SU{0}, it is shown that there is a pointed neighbourhood S0 of 0 such that (S0, S) is A1-convex. It follows from the results of this paper that many expansions of analytic functions in terms of special functions can be considered as eigenfunction expansions of this kind.