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

Abstract

We study linear transformations $T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]$ of the form $T[x^n]=P_n(x)$ where $\{P_n(x)\}$ is a real orthogonal polynomial system. With $T=\sum \tfrac{Q_k(x)}{k!}D^k$, 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 $\{P_n(x)\}$ is an orthogonal system is when the $P_n(x)$ 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.