On some strange summation formulas
The title of this paper is appropriate. The only thing missing in the title is some description of the strangeness, and where it comes from. Some of the series come from integrals involving Bessel functions used in connection with Fourier series. Others come from infinite partial fractions, or from a polynomial identity which is a consequence of Lagrange interpolation. In other words, the identities needed to be found by a wizzard. Some related ones were found by Ramanujan, and most of these were found by Gasper and a computer algebra system. The proofs are clever, and use elementary tools in surprising ways. In the standard form as followed by Ramanujan, a special case will be mentioned here. It is striking, but not illuminating \[ \sum^ \infty_{n=1} n^{-2} \cos \pi \bigl[ n+\sqrt {n^ 2+1} \bigr]^{-1}=\pi^ 2/12. \] The first identity on this paper contains a one parameter extension of a result equivalent to this. For a second value of the parameter, the more general identity reduces to Euler's sum of \(\sum^ \infty_{n=1} n^{-2}\). For some light on where these identities live, see the paper.
scientific article; zbMATH DE number 561182