Transformation and summation formulas for basic hypergeometric series associated with the circumference ratio

Abstract

In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues of Ramanujan's three series for 1/$\pi$ and other eleven nice $\pi$-formulas. Meanwhile, $q$-analogues of three beautiful series for $\zeta (3)$ are also given in the same way.

Some formulas of the paper has been given in a published paper