Some quadratic transformations of basic hypergeometric series and identities in Ramanujan's `lost' note book (Q1110692)

An example of the results obtained is the transformation \[ _ 5\phi_ 4\left[ \begin{matrix} a^ 2,b^ 2,q,x,-x;\\ ab\sqrt{q},-ab\sqrt{q},c,x^ 2q/c;\end{matrix} q;q\right]=_ 4\phi_ 3\left[ \begin{matrix} a^ 2,b^ 2,q^ 2,x^ 2;\\ a^ 2b^ 2q,cq,x^ 2q^ 2/c;\end{matrix} q^ 2;q^ 2\right]. \] Proofs, which proceed by series manipulations and utilization of known results, are merely indicated. The reader's own reconstruction of the proofs is not facilitated by the occurrence of misprints and frequent changes in meanings of symbols.