A partial-sum deformationfor a family of orthogonal polynomials

There are several questions one may ask about polynomials q m ( x ) = q m ( x ; t ) = ∑ n = 0 m t n p n ( x ) attached to a family of orthogonal polynomials { p n ( x ) } n ≥ 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 .