论文标题
用于查找具有多项式系数的线性复发的确定和溶液的段落基础方法
The Factorial-Basis Method for Finding Definite-Sum Solutions of Linear Recurrences With Polynomial Coefficients
论文作者
论文摘要
找到线性复发的非零解决方案的问题$ ly = 0 $，其中$ y $具有确定的超几何总和的形式，与[14] [14] [sec。 8]，现在已经开放了三十年。在这里，我们提出了一种算法（在sagemath套件中实现），鉴于这种复发性和准三角形的，与移位的阶乘基础$ \ mathcal {b} = \ langle p_k（n）\ rangle_ {k = 0}特征零的$ \ mathbb {k} $，计算系数序列$ c = \ langle c_k \ rangle_ {k = 0}^\ infty $ y_n = y_n = \ sum_ = \ sum_ {k = 0} $ \ MATHCAL {B} $，在\ Mathbb {n} $中的每个$ n \）的正确终端总和。更一般而言，如果$ \ Mathcal {b} $是$ m $ - 在\ Mathbb {n} $中的某些$ m \中，我们的算法计算了系数$ c $ C $的$ M $ m $ sections of $ m $复发的系统。如果可以找到该系统的明确非零解决方案，我们将获得$ ly = 0 $的显式非零解决方案。
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 $Ly = 0$.
