Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations

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.