Algebraic Transformations of ₃F₂

Abstract

Let E(x) be the third order differential equation ₃F₂ satisfied by a generalized hypergeometric function ₃F₂ (a₀, a₁, a₂; b₁, b₂; x). Under some assumptions which guarantee that E(x) is irreducible and has no logarithmic solutions, we determine all the algebraic transformations E(x) = θ(x)E′(φ(x)), where E′(z) is a Fuchsian differential equation on P¹, φ(x) a rational function, and θ(x) a finite product of complex powers of rational functions. We find E′(z) also turns out to be a ₃E₂.