Stieltjes Interlacing of Zeros of Jacobi Polynomials from Different Sequences

A theorem of Stieltjes proves that, given any sequence {pn}n=0 of orthogonal polynomials, there is at least one zero of pn between any two consecutive zeros of pk if k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends, under certain conditions, to the zeros of Jacobi polynomials from different sequences. In particular, we prove that the zeros of P n+1 interlace with the zeros of P α+k,β n−1 and with the zeros of P α,β+k n−1 for k ∈ {1, 2, 3, 4} as well as with the zeros of P α+t,β+k n−1 for t, k ∈ {1, 2}; and, in each case, we identify a point that completes the interlacing process. More generally, we prove that the zeros of the kth derivative of P n , together with the zeros of an associated polynomial of degree k, interlace with the zeros of P n+1, k, n ∈ N, k < n.