Diagonal Differential Operators

We explore differential operators, $T$, that diagonalize on a simple basis, $\{B_n(x)\}_{n=0}^\infty$, with respect to some sequence of real numbers, $\{a_n\}_{n=0}^\infty$, and sequence of polynomials, $\{Q_k(x)\}_{k=0}^\infty$, as in $T[B_n(x)]:=\left(\sum_{k=0}^\infty Q_k(x) D^k\right)B_n(x)=a_n B_n(x)$ for every $n\in\mathbb{N}_0$. We discover new relationships between the sequence, $\{Q_k(x)\}_{k=0}^\infty$, and the sequence, $\{a_n\}_{n=0}^\infty$. We find new relationships between polynomial interpolated eigenvalues and the sequence, $\{\deg(Q_k(x))\}_{k=0}^\infty$.