On the Interlacing of Zeros of Linear Combinations of Jacobi Polynomials from Different Sequences

Abstract

We investigate the interlacing of the zeros of linear combinations p_n +aq_m with the zeros of the components p_n and q_m , where {p_n }^∞ _n=0 and {q _m }^∞ _m=0 are different sequences of Jacobi polynomials. The results we prove hold when p_n and q_m are Jacobi polynomials P _n ^(α,β)(x) and P _m ^{(α′,β′)}(x) for certain values of α′ and β′ with m = n or m = n – 1. Numerical counterexamples are given in situations where interlacing fails to occur. We also show that the zeros of the linear combination p _n + aq _m interlace with the zeros of some Jacobi polynomials besides the components of the linear combination.