Properties of the zeros of generalized basic hypergeometric polynomials

We define the generalized basic hypergeometric polynomial of degree $N \geq 1$ in terms of the generalized basic hypergeometric function, which depends on (arbitrary, generic, possibly complex) parameters $q\mathrm{\ne }1$, the $r\ge 0$ parameters ${\alpha }_j$ and the $s\ge 0$ parameters ${\beta }_k$. In this paper we obtain a set of $N$ nonlinear algebraic equations satisfied by the $N$ zeros $\zeta _{n}\equiv \zeta _{n}\left( \underline{\alpha },\underline{\beta };q;N\right) $ofthispolynomial\mathrm{.}Wemoreoveridentifyan$\left( N\times N\right) $-matrix$\underline{M}\equiv \underline{M}\left( \underline{\alpha },\underline{\beta };\underline{\zeta };q;N\right) $featuringthe$N$eigenvalues$\mu {n}=-q^{\left( s-r\right) \left( N-n\right) }\left(q^{-n}-1\right) ~\prod\limits{j=1}^{r}\left( \alpha _{j}~q^{N-n}-1\right)$,where$n=1,2,…,N.$These$N$ eigenvalues depend only on the $r$ parameters ${\alpha }_j$ (besides $q$ and $N$), implying that the $(N\times N)$-matrix $\underset{‾}{M}$ is isospectral for variations of the $s$ parameters ${\beta }_k$; and they clearly are rational numbers if $q$ and the $r$ parameters ${\alpha }_j$ are themselves rational numbers: a nontrivial Diophantine property.