线性分数递归：周期性和可积性
Linear Fractional Recurrences: Periodicities and Integrability
摘要：
我们考虑$z_{n+k} = A(z)/B(z)$形式的k步递归，其中A和B是$z_n, z_{n+1}, ..., z_{n+k-1}$的线性函数，我们称之为k步线性分式递归。本文的第一个定理证明了对每个k，存在k步线性分式递归，它们的周期为4k。在这类递归中，还有所谓的莱尼斯过程，其形式为$A(z)/B(z) = (a +z_{n+1} + z_{n+2} + ... + z_{n+k-1})/z_n$。第二个定理证明了Lyness过程具有二次增长性。Lyness过程是可积的，我们讨论它的已知积分。
英文摘要：
We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step linear fractional recurrences. The first Theorem in this paper shows that for each k there are k-step linear fractional recurrences which are periodic of period 4k. Among this class of recurrences, there is also the so-called Lyness process, which has the form $A(z)/B(z) = (a +z_{n+1} + z_{n+2} + ... + z_{n+k-1})/z_n$. The second Theorem shows that the Lyness process has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.
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