Double Lowering Operators on Polynomials

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