The method of telescoping continued fractions

Abstract

We give an approach to discover continued fractions for series of the form $$\sum _{k=0}^{\infty }\frac{{\epsilon }^k}{(x+k{)}^s},$$ where $\epsilon =\pm 1$. 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.

20 pages, comments solicited