Properties of the zeros of generalized hypergeometric polynomials

Abstract

We define the generalized hypergeometric polynomial of degree N in terms of the generalized hypergeometric function that depends on p parameters a_1, ..., a_p and q parameters b_1, ..., b_q. The parameters are "generic", possibly complex, numbers. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros z_n of this polynomial. We moreover manufacture an NxN matrix L in terms of the 1+p+q parameters N, a_j, b_k characterizing this polynomial, and of its N zeros z_n. We show that the matrix L features N eigenvalues that depend only on the q parameters b_k, implying that this matrix is isospectral for the variations of the p parameters a_j. These eigenvalues are integer (or rational) numbers if the q parameters b_k are themselves integer (or rational) numbers: a nontrivial diophantine property.