求多项式系数线性递推方程组的幂和解的阶乘基方法
The Factorial-Basis Method for Finding Definite-Sum Solutions of Linear Recurrences With Polynomial Coefficients
摘要：
找到具有多项式系数的线性递归$Ly = 0$的非零解的问题，其中$y$具有确定的超几何和的形式，与[14][Sec. [8]至今已开放30年。在这里，我们提出了一个算法（在SageMath软件包中实现），给定这样的递归和特征为零的域$\mathbb{K}$上的多项式空间$\mathbb{K}[n]$的准三角移位兼容阶乘基$\mathcal{B} = \langle P_k(n)\rangle_{k=0}^\infty$，计算解$y_n = \sum_{k=0}^\infty c_kP_k(n)$的系数序列$c = \langle c_k\rangle_{k=0}^\infty$满足的递归（其中，由于$\mathcal{B}$的准三角形性，右侧的和对于每个$n \in \mathbb{N}$终止）。更一般地说，如果$\mathcal{B}$是$m$ -筛选的一些$m \in \mathbb{N}$，我们的算法计算系统的$m$递归满足的$m$ -部分的系数序列$c$。如果能找到该方程组的显式非零解，则得到了$Ly = 0$的显式非零解。
英文摘要：
The problem of finding a nonzero solution of a linear recurrence $Ly = 0$ with polynomial coefficients where $y$ has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of [14][Sec. 8], has now been open for three decades. Here we present an algorithm (implemented in a SageMath package) which, given such a recurrence and a quasi-triangular, shift-compatible factorial basis $\mathcal{B} = \langle P_k(n)\rangle_{k=0}^\infty$ of the polynomial space $\mathbb{K}[n]$ over a field $\mathbb{K}$ of characteristic zero, computes a recurrence satisfied by the coefficient sequence $c = \langle c_k\rangle_{k=0}^\infty$ of the solution $y_n = \sum_{k=0}^\infty c_kP_k(n)$ (where, thanks to the quasi-triangularity of $\mathcal{B}$, the sum on the right terminates for each $n \in \mathbb{N}$). More generally, if $\mathcal{B}$ is $m$-sieved for some $m \in \mathbb{N}$, our algorithm computes a system of $m$ recurrences satisfied by the $m$-sections of the coefficient sequence $c$. If an explicit nonzero solution of this system can be found, we obtain an explicit nonzero solution of $Ly = 0$.
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