On transformations of basic bilateral series and their applications (Q798815)

By series manipulations and utilization of known results the author establishes some transformations involving basic bilateral series; one of these transformations reads, \[ \begin{multlined} \sum^{\infty}_{r=- \infty}\frac{(1-aq^{2r})[x^ 3;q^ 3]_ r[y^ 3;q^ 3]_ r}{(1- a)[a^ 3q^ 3/x^ 3;q^ 3]_ r[a^ 3q^ 3/y^ 3;q^ 3]_ r}\frac{a^{5r}q^{2r(r+1)}}{x^{3r}y^{3r}}= \\ =\prod \left[ \begin{matrix} a^3,q^ 3,a^ 3q^ 3/x^ 3y^ 3;\\ q^ 3,a^ 3q^ 3/x^ 3,a^ 3q^ 3/y^ 3;\end{matrix} q^ 3 \right]\prod [aq,q,q/a;q]\times \sum^{\infty}_{r=-\infty}\frac{[x^ 3;q^ 3]_ r[y^ 3;q^ 3]_ r}{[a^ 3;q^ 3]_{2r}}\frac{a^{3r}q^{3r}}{x^{3r}y^{3r}}. \end{multlined} \] By consideration of special and limiting cases, a number of interesting results (summation and product formulae, identities of Rogers-Ramanujan type, continued fractions) are derived. Finally, the author gives \[ \begin{multlined} _ 8\Phi_ 7\left[ \begin{matrix} -z,iq\sqrt{z},-iq\sqrt{z},-zq/a^ 2,a,-a,a\sqrt{q},-a\sqrt{q};\\ i\sqrt{z},-i\sqrt{z},a^ 2,- zq/a,zq/a,z\sqrt{q}/a,-z\sqrt{q}/a; \end{matrix} q;-\frac{z}{a^ 2} \right]= \\ =\prod \left[ \begin{matrix} -zq,z^ 2q/a^ 4;\\ -z/a^ 2,z^ 2q/a^ 2;\end{matrix} q \right], \quad| z/a^ 2| <1, \end{multlined} \] and one more summation formula of a similar appearance.