Multisection method and further formulae for \(\pi\)
The author considers an elementary approach for deriving infinite series expression of~\(\pi\) in the form \[ \sum_{n=0}^\infty\frac{P_m(n)\alpha^n}{\binom{2mn}{mn}}, \] where \(\alpha\)~is a rational number and \(P_m(n)\)~a polynomial of degree \(m\) in~\(n\); the cases covered include \(m=4,8,12,16,24\) with the announced possibility to extend the argument to~\(m\) being an arbitrary positive power of~2. A systematic search and proof of such identities was done previously in [\textit{G.~Almkvist, C.~Krattenthaler}, and \textit{J.~Petersson}, Exp. Math. 12, No. 4, 441--456 (2003; Zbl 1161.11419)].
scientific article; zbMATH DE number 5300047