Functions satisfying Poincaré's multiplication formula

Abstract

Poincaré proved existence of meromorphic function solution of a certain kind of system of multiplication formulae and claimed that the class of those functions is new. In this paper we exemplify Poincaré's meromorphic functions being truly new in a rigorous sense. We prove that those functions cannot be expressed rationally (nor algebraically) by solutions of linear difference equations, the exponential function $\mathrm{e}^{x}$, the trigonometric functions $\cos x$ and $\sin x$, the Weierstrass function $\wp(x)$ and any other functions satisfying first order algebraic difference equations, where the transforming operator of the difference equations is one sending $y(x)$ to $y(2x)$, not to $y(x+1)$.