Real-Root Preserving Differential Operator Representations of Orthogonal Polynomials

In this paper, we study linear transformations of the form $T[x^n]=P_n(x)$ where $\{P_n(x)\}$ is an orthogonal polynomial system. Of particular interest is understanding when these operators preserve real-rootedness in polynomials. It is known that when the $P_n(x)$ are the Hermite polynomials or standard Laguerre polynomials, the transformation $T$ has this property. It is also known that the transformation $T[x^n]=H_n^{\alpha}(x)$, where $H_n^{\alpha}(x)$ is the $n$th generalized Hermite Polynomial with real parameter $\alpha$, has the differential operator representation $T[x^n]=e^{-\frac{\alpha}{2}D^2}x^n$. The main result of this paper is to prove that a differential operator of the form $\sum_{k=0}^\infty \frac{\gamma_k}{k!} D^k$ induces a system of monic orthogonal polynomials if and only if $\sum_{k=0}^\infty \frac{\gamma_k}{k!} D^k=\gamma_0e^{-\frac{ \alpha}{2}D^2-\beta D}$ where $\gamma_0,\alpha,\beta \in \mathbb{C}$ and $\alpha,\gamma_0 \neq 0$. This operator will produce a shifted set of generalized Hermite polynomials when $\alpha \in \mathbb{R}$. We also express the transformation from the standard basis to the standard Laguerre basis, $T[x^n]=L_n(x)$ as a differential operator of the form $\sum_{k=0}^\infty \frac{p_k(x)}{k!} D^k$ where the $p_k$ are polynomials, an identity that has not previously been shown.