The method of telescoping continued fractions

Source abstract

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.