A statement of Weierstrass is known for meromorphic functions which admit an algebraic addition theorem. We give its precise formulation and prove it complex analytically. In fact, we show that if is a non-degenerate algebraic function field in variables over which admits an algebraic addition theorem, then any is a rational function of some coordinate functions and abelian functions of other variables.
"A statement of Weierstrass on meromorphic functions which admit an algebraic addition theorem." J. Math. Soc. Japan 57 (3) 709 - 723, July, 2005. https://doi.org/10.2969/jmsj/1158241931