Properties of the zeros of generalized basic hypergeometric polynomials

Abstract

We define the generalized basic hypergeometric polynomial of degree $N\ge 1$ in terms of the generalized basic hypergeometric function, which depends on (arbitrary, generic, possibly complex) parameters $q\ne 1$, 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 ${\zeta }_n\equiv {\zeta }_n\left(\underline{\alpha },\underline{\beta };q;N\right)$ of this polynomial. We moreover identify an $\left(N\times N\right)$-matrix $\underline{M}\equiv \underline{M}\left(\underline{\alpha },\underline{\beta };\underline{\zeta };q;N\right)$ featuring the $N$ eigenvalues ${\mu }_n=-q^{\left(s-r\right)\left(N-n\right)}\left(q^{-n}-1\right)\prod _{j=1}^r\left({\alpha }_j{q}^{N-n}-1\right)$, where $n=1,2,...,N.$ These $N$ eigenvalues depend only on the $r$ parameters ${\alpha }_j$ (besides $q$ and $N$), implying that the $\left(N\times N\right)$-matrix $\underline{M}$ 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.