Ramanujan Series With a Shift

Abstract

We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_{0}$, we call the resulting series ‘Ramanujan series with the shift $x_{0}$’. Then we relate these shifted series to some $q$-series and solve the case of level $4$ with the shift $x_{0}=1/2$. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels $\ell =1,2,3$ and the shift $x_{0}=1/2$.

On Some Classes of Series Representations for $1/\pi$ and $\pi^2$

“…It should be remarked that before Ramanujan in 1859 G. Bauer [1], and in 1905 W. L. Glaisher [10] had given series representations for 1/π. The studies on Ramanujan-like series for 1/π are continuing intensively today, too and recently, many new series of this type have been published, see for example [7,8,[11][12][13][14][15][16][17][18][19][20][21][22][23]. The aim of this paper is to derive new classes of series representations for 1/π and π 2 by using the W Z-method.…”