递归关系的主解与数论中的无理数问题
Principal solutions of recurrence relations and irrationality questions in number theory
摘要：
本文将非共轭线性递推关系理论应用于数论中无理量的研究。特别是，对于一个无理数与解决方案的三项线性递归关系，我们表明，存在一个四项线性递归关系的解决方案，使我们能够证明，这个数字是一个二次无理数当且仅当四项递归关系有一个主解的某种类型。结果被推广到高阶递归关系，超越准则也可以用这些主解来表示。当应用到权力的情况下的$ζ(3)$它是不知道是否相应的四项递归关系没有或没有这样一个主要的解决方案，但该方法确实产生新的系列扩展的权力的$ζ(3)$和$ζ(2)$在阿佩里的经典序列。
英文摘要：
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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