Diagonal Differential Operators

Abstract

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, $\{°(Q_k(x)){\}}_{k=0}^{\infty }$.

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References in corpus (6)

• Polynomials, roots, and interlacing
• Hermite multiplier sequences and their associated operators
• Abstract "hypergeometric" orthogonal polynomials
• Quadratic Hyperbolicity Preservers & Multiplier Sequences
• A characterization of multiplier sequences for generalized Laguerre bases
• Polynomial solutions for a class of second-order linear differential equations