Algebraic Independence of Solutions of First-Order Rational Difference Equations

Abstract

We prove a theorem on algebraic independence of solutions of first order rational difference equations. By the theorem, we are able to prove algebraic independence of x, the exponential function e^x and the Weierstrass function ${\wp(x)}$ over ${\mathbb{C}}$ only by seeing degrees of polynomials associated with their double angle formulas. As a corollary, we obtain a result on unsolvability of a first-order rational difference equation by solutions of other first-order rational difference equations, which implies its irreducibility. Additionally, we introduce some applications to algebraic independence of functions f(x), f(x ²), . . . , f(x ⁿ).