A partial-sum deformation for a family of orthogonal polynomials

There are several questions one may ask about polynomials qₘ(x)=qₘ(x;t)=∑ₙ₌₀ᵐtᵐpₙ(x) attached to a family of orthogonal polynomials ₙ(x)≥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 qₘ(x;t) in the case of varying real parameter t.