On some continued fraction expansions for ratios of \(_2\psi_2\) (Q2751683)

The authors prove four unusual continued fraction expansions, three for the quotient of contiguous basic hypergeometric \(_2\phi_1\)-series, and one for the quotient of contiguous bilateral basic hypergeometric \(_2\psi_2\)-series, in which the definition for the coefficients of the continued fraction splits in three cases. For example, Theorem~3.2 says that NEWLINE\[NEWLINE \frac {\scriptstyle_2\psi_2\left[\begin{smallmatrix} a,bq\\cq,d\end{smallmatrix};xq\right]} {\scriptstyle_2\psi_2\left[\begin{smallmatrix} a,b\\c,d\end{smallmatrix};xq\right]}= \frac {\frac {(1-c)((d/q)-b)} {(1-b)}} {E_0+} \frac {F_0} {G_0+}\frac {H_0} {I_0+}\frac {J_0} {E_1+} \frac {F_1} {G_1+}\frac {H_1} {I_1+}\frac {J_1} {E_2+}\cdots,NEWLINE\]NEWLINE with explicitly given sequences \(E_i,F_i,G_i,H_i,I_i,J_i\). They obtain their results by cleverly combining contiguous relations. Several interesting special and limiting cases are also listed.NEWLINENEWLINENEWLINEReviewer's remark: In fact, all of the theorems are stated for \(_2\psi_2\)-series. However, Theorems~3.1, 3.3, 3.4 have \(_2\psi_2\)-series where one of the bottom parameters, \(d\), is a power \(q^m\) with \(m\) a positive integer. It is easy to check that all these theorems are equivalent to the corresponding ones with \(d=q\), in which case the \(_2\psi_2\)-series reduce to \(_2\phi_1\)-series.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00018].