On a functional equation of A. Hurwitz (Q1842218)

The paper deals with the equation \((*)\) \(G'(z) = 2 \pi ie^{2 \pi iz} G(z + 1)\) and its adjoint equation \(F'(z) = - 2 \pi ie^{2 \pi iz} F(z - 1)\) in the complex domain. It is shown that \(G_ 0(z) = \int e^{-2 \pi iuz + \pi iu^ 2} \Gamma (u) du\) and \(F_ 0(z) = \int e^{2 \pi izu - \pi iu^ 2} \Gamma^{-1} (1 + u) du\) with integration over suitable contours are special entire solutions. \(\widetilde G(z) = \exp (e^{2 \pi iz})\), \(G_ n(z) = G_ 0 (z + n)\) and \(\widetilde F(z) = \exp (- e^{2 \pi iz})\), \(F_ n(z) = F_ 0 (z + n)\) with \(n \in \mathbb{Z}\) are further solutions. The general entire solution of \((*)\) can be represented as \(G(z) = c \widetilde G(z) + \sum^{+ \infty}_{- \infty} c_ n G_ 0 (z + n)\). Using the inner product \(\{F,G\} = F(z) G(z) - 2 \pi i_{z-1} \int^ ze^{2 \pi ix} F(x) G(x + 1) dx\), which is constant for solutions of the foregoing equations, and the fact that the foregoing solutions form a biorthogonal system, the coefficients are the Fourier coefficients \(c = \{\widetilde F, G\}\), \(c_ n = (i/(2 \pi)) \{F_ n, G\}\). There are given asymptotic expansions of \(F_ 0 (z)\) and \(G_ 0 (z)\) for \(z \to \infty\) in different sectors, and estimates for \(c_ n\). Moreover, relations to automorphic functions are pointed out.