  We study linear transformations $T \colon \mathbb{R}[x] \to \mathbb{R}[x]$ of
the form $T[x^n]=P_n(x)$ where $\{P_n(x)\}$ is a real orthogonal polynomial
system. Such transformations that preserve or shrink the location of the
complex zeros of polynomials is a recent object of study, motivated by the
Riemann Hypothesis. In particular, we are interested in linear transformations
that map polynomials with all real zeros to polynomials with all real zeros. It
is well known that any transformation $T \colon \mathbb{C}[x] \rightarrow
\mathbb{C}[x]$ has a differential operator representation $T = \sum_{k =
0}^\infty \frac{Q_k(x)}{k!} D^k$. Motivated by the work of Chasse
\cite{Chasse-PhD-2011}, Forg\'acs, and Piotrowski
\cite{Forgacs-Piotrowski-Hermite-2015}, we seek to understand the behavior of
the transformation $T$ by studying the roots of the $Q_k(x)$. We prove four
main things. First, we show that the only case where the $Q_k(x)$ are constant
and $\{P_n(x)\}$ are an orthogonal system is that when the $P_n$ form a shifted
set of generalized probabilist Hermite polynomials. Second, we show that the
coefficient polynomials $Q_k(x)$ have real roots when the $P_n(x)$ are the
physicist Hermite polynomials or the Laguerre polynomials. Next, we show that
in these cases, the roots of successive polynomials strictly interlace, a
property that has not yet been studied for coefficient polynomials. We conclude
by discussing the Chebyshev and Legendre polynomials, proving a conjecture of
Chasse, and presenting several open problems.
