I first learned of the below identity from MathWorld and the works of Ramanujan, but it's completely crazy with polygammas and Laplace transforms of hyperbolic trig. It seems weird that the Laplace Transform of $\frac{t}{\cosh{t}}$ at $s=\sqrt5$ should yield such an interesting continued fraction; why this function and why that value of $s$?
The connection between the integral and the polygammas makes sense by integrating with power series. The connection to the continued fraction completely evades me, however. How does one go about deriving and/or verifying the veracity of this identity?
