Author: Ratcliffe, James Wells
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.
Copyright is held by the author.
The author granted permission for the file to be printed and for the text to be copied and pasted.
Supervisor or Senior Supervisor
Thesis advisor: Bell, Jason
Member of collection