Ramanujan series with a shift

TL;DR

This paper extends Ramanujan series by introducing a variable shift, relates these to q-series, and explores specific cases and patterns, aiming to deepen understanding of their structure and potential proofs.

Contribution

It introduces a shifted version of Ramanujan series, connects them to q-series, and investigates specific level cases and patterns for potential proofs.

Findings

Relation of shifted Ramanujan series to q-series.

Solution for level 4 with shift 1/2.

Indication of methods to prove observed patterns.

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 $x_0=1/2$.