Interlacing theorems for the zeros of some orthogonal polynomials from different sequences

摘要

We study the interlacing properties of the zeros of orthogonal polynomials p_n and r_m, m = n or n - 1 where {p_n}_(n=1)~∞ and {r_m}_(m=1)~∞ are different sequences of orthogonal polynomials. The results obtained extend a conjecture by Askey, that the zeros of Jacobi polynomials p_n = P_n~(α,β) and r_n = p_n~(γ,β) interlace when α < γ ≤ α + 2, showing that the conjecture is true not only for Jacobi polynomials but also holds for Meixner, Meixner-Pollaczek, Krawtchouk and Hahn polynomials with continuously shifted parameters. Numerical examples are given to illustrate cases where the zeros do not separate each other.