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.