Properties of the zeros of generalized basic hypergeometric polynomials

الأصل · EN

We define the generalized basic hypergeometric polynomial of degree N ≥ 1 in terms of the generalized basic hypergeometric function, which depends on (arbitrary, generic, possibly complex) parameters q ≠ 1, the r ≥ 0 parameters αⱼ and the s ≥ 0 parameters βₖ. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros ζₙ≡ ζₙ(α,β;q;N) of this polynomial. We moreover identify an (N× N) -matrix M≡ M(α,β;ζ;q;N) featuring the N eigenvalues μₙ=-q(s-r) (N-n)(q⁻ⁿ-1) ∏ⱼ₌₁ʳ(αⱼ qⁿ⁻ⁿ-1), where n=1,2,...,N. These N eigenvalues depend only on the r parameters αⱼ (besides q and N), implying that the (N× N) -matrix M is isospectral for variations of the s parameters βₖ; and they clearly are rational numbers if q and the r parameters αⱼ are themselves rational numbers: a nontrivial Diophantine property.