A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.

翻译：称一个序列为$C$有限序列，若其满足常系数线性递推关系。本文研究满足$C$有限系数线性递推的序列。近期研究表明，此类$C^2$有限序列与$C$有限序列具有相似的闭包性质，特别地，它们构成一个差分环。本文提出了实现$C^2$有限序列闭包性质的新技术，这些方法使我们能够推导出此前未知的阶界，并进一步揭示这些计算的有效性。研究结果基于代数数的指数格，我们给出了可用于计算此类格基的迭代算法。