Interlacing of zeros from different sequences of Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials

In this paper we consider interlacing of the zeros of polynomials from different sequences $\{p_n\}$ and $\{g_n\}$. In our main result we consider a mixed recurrence equation necessary for existence of a linear term $(x-A)$ so that the $(n+1)$ zeros of $(x-A)g_n(x)$ interlace with the $n$ zeros of $p_n$. We apply our result to Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials to obtain new interlacing results for the zeros of polynomials of the same degree from different polynomial sequences.