Separating zeros of polynomials using an added interlacing point

Following a systematic analysis of existing results, we investigate when complete interlacing between the zeros of distinct polynomial sequences, $\{\mathcal{P}_n\}$ and $\{\mathcal{G}_n\}$ can be achieved by using a naturally arising extra point. Specifically, we analyse several general mixed recurrence relations that ensure the $n+1$ zeros of the polynomial $(x-E)\mathcal{P}_n(x)$ interlace with the $k$ zeros of $\mathcal{G}_k$, where $k=n$ or $n+1$. In addition, we show that imposing specific conditions on the extra point $E$ yields full interlacing between the zeros of $\mathcal{P}_n$ and $\mathcal{G}_k$ for a suitable choice of $n$. The approach provides a consolidated framework broadly applicable to both orthogonal and non-orthogonal polynomials and we illustrate this with new interlacing results for zeros of Krawtchouk, Meixner, and Narayana polynomials. We also illustrate that this general approach can be used to recover and refine existing results regarding the complete interlacing of zeros for classical Jacobi and Laguerre polynomials.