Undecidable arithmetic properties of solutions of Fredholm integral equations

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 (alpha) is transcendental or algebraic if f(z) is an E -function and alpha is an element of & nbsp;((Q*)over bar) 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) is an element of Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6]. (C) 2021 Elsevier Inc. All rights reserved.