$q$-series reciprocities and further $\pi$-formulae

Abstract

By examining reciprocal relations of basic well-poised, quadratic and cubic series, we establish $q$-analogues of three infinite series for $1/\pi^2$ due to Guillera (2003) and $\lambda$-parameter extensions of three infinite series for $1/\pi$ due to Ramanujan (1914). Several further infinite series identities of Ramanujan-type are also derived as consequences.