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)(infinity) and {r(m)}(m=1)(infinity) are different sequences of orthogonal polynomials. The results obtained extend a conjecture by Askey, that the zeros of Jacobi polynomials p(n) = P-n((alpha,beta)) and r(n) = P-n((gamma,beta)) n interlace when alpha < gamma <= alpha + 2, showing that the conjecture is true not only for Jacobi polynomials but also holds for Meixner, Meixner-Pollaczek, Krawtchouk and Hahn polynornials with continuously shifted parameters. Numerical examples are given to illustrate cases where the zeros do not separate each other