Principal Solutions of Recurrence Relations and Irrationality Questions in Number Theory

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of three-term linear recurrence relations we show that there exists a four-term linear recurrence relation whose solutions allow us to show that the number is a quadratic irrational if and only if the four-term recurrence relation has a principal solution of a certain type. The result is extended to higher order recurrence relations and a transcendence criterion can also be formulated in terms of these principal solutions. When applied to the situation of powers of ζ(3) it is not known whether the corresponding four term recurrence relation does or does not have such a principal solution, however the method does generate new series expansions of powers of ζ(3) and ζ(2) in terms of Apéry’s now classic sequences.