Hypertranscendence of the multiple sine function for a complex period (Q2418809)

Let \(\omega_1,\dots,\omega_r\in\mathbb{C}\) all lie on the same side of some straight line through the origin. Put \(\omega= (\omega_1,\dots, \omega_r)\). The multiple sine function is fined by \textit{N. Kurokawa} and \textit{S.-y. Koyama} [Forum Math. 15, No. 6, 839--876 (2003; Zbl 1065.11065)] \(\text{Sin}_r(x,\omega)= \Gamma_r(\omega_1+\cdots+ \omega_r-x,\omega)^{(-1)^r}\cdot\Gamma_r(x,\omega)^{-1}\), where \(\Gamma_r(x,\omega)= \exp(\frac{\partial}{\partial s}\,\zeta_r(s,x,\omega)|_{s=0})\) and \(\zeta_r(s,x,\omega)= \sum_{n_1,\dots, n_r\ge 0}(x+ n_1\omega_1+\cdots+ n_r\omega_r)^{-s}\). An old theorem of \textit{O. Hölder} [Gött. Nachr. 1887, 662--676 (1887; JFM 19.0378.01)] states that the classical differential equation. The author now proves that if there exists a non-real element in the set \(\{\omega_j/\omega_i: 1\le i<j\le r\}\), \(r\ge 2\), then the function \(\text{Sin}_r(x,\omega)\) is hypertranscendental. A similar property for Appell's \(\sigma\)-function is proved, too.