A certain series associated with Catalan's constant
The series \(s(r)=\sum^\infty_{k=0} {{2k\choose k}^2\over (k+r) 16^k}\) was studied by Ramanujan in the cases for which \(r=n\) is a positive integer, and when \(r=n+ \frac 12\) is a half integer. The purpose of the article under review is to study this series in the case when \(r=-n\) is a negative integer, in which case the term \(k=-r\) is omitted from the series defining \(S(r)\). Using contour integration, a formula for \(S(r)\) is given in terms of \(S(\frac 12-r)\) and the harmonic numbers.
scientific article; zbMATH DE number 1827613