Question: For a sequence of real polynomials $p_k(z)$, if the zeros of each of them are real and distinct, and those zeros are totally interlacing. Will the zeros of linear combination of $p_k(z)$ such as:
${\displaystyle\sum c_{k}p_k(z) }$
all be real ? Here, linear combination coefficient $c_{k}$ are all real.
Interesting question. If you take the sequence of orthogonal polynomials for a certain measure, then the polynomials will have interlacing roots. Note that every polynomial is a linear combination of them. So probably you want positive linear combinations.
Added: In fact, the roots of orthogonal polynomials (for some measure) have the property that between any two roots of $f_m$ there is a root of $f_n$, for any $m$ and $n$. Is that what you mean by strong interlacing?