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

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.