Double Lowering Operators on Polynomial

Recently Sarah Bockting-Conrad introduced the double lowering operator $\psi$ for a tridiagonal pair. Motivated by $\psi$ we consider the following problem about polynomials. Let $\mathbb{F}$ denote an algebraically closed field. Let $x$ denote an indeterminate, and let $\mathbb{F}[x]$ denote the algebra consisting of the polynomials in $x$ that have all coefficients in $\mathbb{F}$. Let $N$ denote a positive integer or $\mathrm{\infty }$. Let $\lbrace a_i\rbrace_{i=0}^{N-1}$,$\lbrace b_i\rbrace_{i=0}^{N-1}$denotescalarsin$\mathbb F$suchthat$\sum_{h=0}^{i-1} a_h \not= \sum_{h=0}^{i-1} b_h$for$1 \leq i \leq N$\mathrm{.}For$0 \leq i \leq N$definepolynomials$\tau_i, \eta_i \in \mathbb F\lbrack x \rbrack$by$\tau_i = \prod_{h=0}^{i-1} (x-a_h)$and$\eta_i = \prod_{h=0}^{i-1} (x-b_h)$\mathrm{.}Let$V$denotethesubspaceof$\mathbb F\lbrack x \rbrack$spannedby$\lbrace x^i\rbrace_{i=0}^N$\mathrm{.}Anelement$\psi \in \operatorname{End}(V)$iscalleddoubleloweringwhenever$\psi \tau_i \in \mathbb F \tau_{i-1}$and$\psi \eta_i \in \mathbb F \eta_{i-1}$for$0 \leq i \leq N$,where$\tau_{-1}=0$and$\eta_{-1}=0$. We give necessary and sufficient conditions on $\{a_i{\}}_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.