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...
Ausführliche Beschreibung

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. Ausführliche Beschreibung