Noncommutative $Q$-Gauss summations for basic hypergeometric series

Let $A$, $B$, and $C$ be noncommutative parameters in a Banach algebra, with $Q$ commuting with each of $A$, $B$, and $C$. Assume that $BC^{-1}A$ commutes with each of $A$, $B$, and $C$, and that

$$\|A^{-1}CB^{-1}\|<1.$$

Write ${}_2\phi_1$ for the noncommutative basic hypergeometric series and use the corresponding infinite noncommutative $Q$-shifted-factorial notation. Noncommutative $Q$-Gauss summations. The type I and type II series should satisfy, respectively,

$${}_2\phi_1\!\left\lceil\begin{matrix}A,B\\ C\end{matrix}\,;Q,A^{-1}CB^{-1}\right\rfloor= \left\lceil\begin{matrix}CB^{-1},A^{-1}C\\ C,A^{-1}CB^{-1}\end{matrix};Q,I\right\rfloor_\infty,$$

and

$${}_2\phi_1\!\left\lfloor\begin{matrix}A,B\\ C\end{matrix}\,;Q,A^{-1}CB^{-1}\right\rceil= \left\lfloor\begin{matrix}CB^{-1}Q,A^{-1}CQ\\ C,A^{-1}CB^{-1}\end{matrix};Q,I\right\rceil_\infty\times (I-CB^{-1})(I-A^{-1}C).$$

These are noncommutative $Q$-analogues of Gauss summation formulas, extending the corresponding classical identities. Their validity is conjectural because they are obtained by taking the termwise limit $n\to\infty$ in a finite summation; convergence and the interchange of limit and summation require justification in each case.