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 ex 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(x2), . . . , f(xn).