Properties of the zeros of generalized hypergeometric polynomials

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