Undecidable arithmetic properties of solutions of Fredholm integral equations

Abstract

A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E-functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f(α) is transcendental or algebraic if f(z) is an E-function and α∈Q‾⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f(z) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f(0)∈Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6].