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

We study linear transformations $$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]$$ 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 behavior transformation T by studying roots $$Q_k(x)$$ . prove four main things. First, show that only case are constant and an system when $$P_n(x)$$ shifted set generalized probabilist Hermite polynomials. Second, coefficient polynomials have physicist or Laguerre Next, in these cases, successive strictly interlace, property has not yet been studied for conclude discussing Chebyshev Legendre polynomials, proving conjecture Chasse, presenting several open problems.