Hypertranscendence of the multiple sine function for a complex period

Abstract

It is known that the multiple sine function for a “rational” period satisfies an algebraic differential equation. However, for a non-“rational” period, the differential algebraicity of the multiple sine function is obscure. In this paper, we prove that, if there exists a non-real element in the set $\{\omega_{j}/\omega_{i}|1\leq i<j\leq r\}$, the multiple sine function $\text{Sin}_{r}(x,(\omega_{1},\cdots,\omega_{r}))$ does not satisfy any algebraic differential equation.