Properties of the zeros of generalized hypergeometric polynomials

Abstract

We define the generalized hypergeometric polynomial of degree N as follows:PN(α1,...,αp;β1,...,βq;z)=∑m=0N[(−N)m(α1)m⋅⋅⋅(αp)mzN−mm!(β1)m⋅⋅⋅(βq)m]=zNFqp+1(−N,α1,...,αp;β1,...,βq;1/z). Here N is an arbitrary positive integer, p and q are arbitrary nonnegative integers, the p+q parameters αj and βk are arbitrary (“generic”, possibly complex) numbers, (α)m is the Pochhammer symbol and Fqp+1(α0,α1,...,αp;β1,...,βq;z) is the generalized hypergeometric function. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros ζn of this polynomial. We moreover manufacture an N×N matrix L̲ in terms of the 1+p+q parameters N, αj, βk characterizing this polynomial, and of its N zeros ζn, and we show that it features the N eigenvalues λm=m∏k=1q(−βk+1−m), m=1,...,N. These N eigenvalues depend only on the q parameters βk, implying that the N×N matrix L̲ is isospectral for variations of the p parameters αj; and they clearly are integer (or rational) numbers if the q parameters βk are themselves integer (or rational) numbers: a nontrivial Diophantine property.