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Principal solutions of recurrence relations and irrationality questions in number theory

Angelo B. Mingarelli
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Abstract

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.

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Angelo B. Mingarelli. 2008-03-27. Principal solutions of recurrence relations and irrationality questions in number theory. https://arxiv.org/abs/math/0608577

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