Sums of rational functions

Given a rational function φ(X) with rational coefficients that is defined at every positive integer, we consider the sum of φ(n) as n runs from 0 to infinity. It is believed that when this sum converges, it converges to either a rational or transcendental number. We prove an analogue of this conjecture over fields of rational functions: Let K be a field and let φ(X) be a rational function with coefficients in K such that φ(0) = 0. Given a positive integer d ≥ 2, we define F(X) to be the sum of φ(X^(d^n)) as n runs from 0 to infinity. If d is not a power of char(K), then F(X) is either a rational function or transcendental over K(X). Our demonstration uses results from the theory of automatic sequences and from commutative algebra.