发表机构
Jawaharlal Nehru University（贾瓦哈拉尔·尼赫鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出望远镜连分数方法，用于发现形如∑ε^k/(x+k)^s的级数的连分数，推广了Ramanujan的连分数，并展示了s=5,7,9,11时的前几项。
AI 中文摘要
我们给出一种方法来发现形如 $$\sum_{k=0}^\infty \frac{\epsilon^k}{(x+k)^s}$$ 的级数的连分数，其中 $\epsilon = \pm 1$。我们找到形如 \begin{equation*} \frac{a_1}{b_1(x)} \fplus \frac{a_2}{b_2(x)} \fplus \fdots \end{equation*} 的连分数，其中 $a_k$ 是常数，$b_k(x)$ 是多项式。我们的技术涉及望远镜连分数。这为 Ramanujan 给出的以下连分数提供了发现方法：$$2\sum_{k=1}^\infty \frac{(-1)^{k+1}}{x+2k-1}, 2\sum_{k=0}^\infty \frac{1}{(x+2k+1)^2}, 2\sum_{k=0}^\infty \frac{(-1)^k}{(x+2k+1)^2}, \sum_{k=1}^\infty \frac{1}{(x+k)^3}, $$ 以及类似形式。我们展示了对于较大的 $s$ 值（包括 $s=5, 7, 9, 11$）以这种方式获得的几个连分数的前几项。它们并不遵循像 Ramanujan 连分数那样简单的模式。
英文摘要
We give an approach to discover continued fractions for series of the form $$\sum_{k=0}^\infty \frac{ε^k}{(x+k)^s},$$ where $ε= \pm 1$. We find a continued fraction of the form \begin{equation*} \frac{a_1}{b_1(x)} \fplus \frac{a_2}{b_2(x)} \fplus \fdots \end{equation*} where $a_k$ are constants and $b_k(x)$ are polynomials. Our technique involves telescoping continued fractions. This provides a discovery approach to continued fractions given by Ramanujan for $$2\sum_{k=1}^\infty \frac{(-1)^{k+1}}{x+2k-1}, 2\sum_{k=0}^\infty \frac{1}{(x+2k+1)^2}, 2\sum_{k=0}^\infty \frac{(-1)^k}{(x+2k+1)^2}, \sum_{k=1}^\infty \frac{1}{(x+k)^3}, $$ and the like. We display the first few terms of several continued fractions obtained in this manner for larger values of $s$, including $s=5, 7, 9, 11$. They do not follow as simple a pattern as Ramanujan's continued fractions.
Comments20 pages, comments solicited
