常非齐次线性系数非齐次二阶递推关系的超越解
Transcendental solution to linear coefficient non-homogeneous second order recurrence relation with constant non-homogenity
摘要：
实数的二阶递推关系在离散时间动力系统和马尔可夫链中有着广泛的应用。递推关系的解完全由前两个初值和递推公式定义。当非齐次项为常数且为负，第一初始值为$a_0=0$时，我们计算了仿射系数非齐次二阶递推关系的最小正解$(a_i)_{i\in\mathbb{N}}$作为$a_1$的函数.我们证明了有理系数导致一个超越数序列。另外，我们证明了当$a_1$变化时，该序列是唯一有界解，收敛到$0$，并给出了在$O(i^{-1})$中的收敛速度.我们评论在最后一节进一步的合理参数的选择，在递归关系，我们做了一个链接，以获得基于计算机的可视化的最小正解$(a_i)_{i\in\mathbb{N}}$的不可能性。
英文摘要：
Second order recurrence relations of real numbers arise form various applications in discrete time dynamical systems as well as in the context on Markov chains. Solutions to the recurrence relations are fully defined by the first two initial values as well as the recurrence formula. We calculate in this work explicitly as a function of $a_1$ the minimal positive solution $(a_i)_{i\in\mathbb{N}}$ to non-homogeneous second order recurrence relation with affine coefficients when the non-homogeneity is constant and negative, and the first initial value equals $a_0=0$. We show that rational coefficients lead to a sequence of transcendental numbers. Additionally, we prove that this sequence is the only bounded solution when varying $a_1$, converges to $0$ and obtain the convergence speed in $O(i^{-1})$. We comment in the last section further on the choice of rational parameters in the recurrence relation and we make a link to the impossibility of obtaining computer based visualizations of the minimal positive solution $(a_i)_{i\in\mathbb{N}}$.
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