A partial-sum deformation for a family of orthogonal polynomials

There are several questions one may ask about polynomials $q_m(x)=q_m(x;t)={\sum }_{n=0}^m{t}^m{p}_n(x)$ attached to a family of orthogonal polynomials $\{p_n(x){\}}_{n\ge 0}$. In this note we draw attention to the naturalness of this partial-sum deformation and related beautiful structures. In particular, we investigate the location and distribution of zeros of $q_m(x;t)$ in the case of varying real parameter $t$.