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

We study linear transformations $$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]$$ T : R [ x ] → R [ x ] of the form $$T[x^n]=P_n(x)$$ T [ x n ] = P n ( x ) where $$\{P_n(x)\}$$ { P n ( x ) } is a real orthogonal polynomial system. With $$T=\sum \tfrac{Q_k(x)}{k!}D^k$$ T = ∑ Q k ( x ) k ! D k , we seek to understand the behavior of the transformation T by studying the roots of the $$Q_k(x)$$ Q k ( x ) . We prove four main things. First, we show that the only case where the $$Q_k(x)$$ Q k ( x ) are constant and $$\{P_n(x)\}$$ { P n ( x ) } is an orthogonal system is when the $$P_n(x)$$ P n ( x ) form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials $$Q_k(x)$$ Q k ( x ) have real roots when the $$P_n(x)$$ 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.