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Double Lowering Operators on Polynomial

Recently Sarah Bockting-Conrad introduced the double lowering operator $ψ$ for a tridiagonal pair. Motivated by $ψ$ we consider the following problem about polynomials. Let $\mathbb F$ denote an algebraically closed field. Let $x$ denote an indeterminate, and let $\mathbb F\lbrack x \rbrack$ denote the algebra consisting of the polynomials in $x$ that have all coefficients in $\mathbb F$. Let $N$ denote a positive integer or $\infty$. Let $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ denote scalars in $\mathbb F$ such that $\sum_{h=0}^{i-1} a_h \not= \sum_{h=0}^{i-1} b_h$ for $1 \leq i \leq N$. For $0 \leq i \leq N$ define polynomials $τ_i, η_i \in \mathbb F\lbrack x \rbrack$ by $τ_i = \prod_{h=0}^{i-1} (x-a_h)$ and $η_i = \prod_{h=0}^{i-1} (x-b_h)$. Let $V$ denote the subspace of $\mathbb F\lbrack x \rbrack$ spanned by $\lbrace x^i\rbrace_{i=0}^N$. An element $ψ\in \operatorname{End}(V)$ is called double lowering whenever $ψτ_i \in \mathbb F τ_{i-1}$ and $ψη_i \in \mathbb F η_{i-1}$ for $0 \leq i \leq N$, where $τ_{-1}=0$ and $η_{-1}=0$. We give necessary and sufficient conditions on $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.