Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians

We study the difference analog of quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space quasi-exponentials $W=\langle \alpha_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $\alpha_{i}\in{\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider formal conjugate $\check{S}^{\dagger}_{W}$ $\check{S}_{W}$ satisfying $\widehat{S} =\check{S}_{W}S_{W}$. Here, $S_{W}$ is linear order $\dim W$ annihilating $W$, $\widehat{S}$ constant coefficients depending on $\alpha_{i}$ $\deg p_{ij}(x)$ only. construct dimension $\operatorname{ord} \check{S}^{\dagger}_{W}$, which annihilated by describe its basis discrete exponents. also similar construction for operators associated spaces quasi-polynomials, combinations functions form $x^{z}q(x)$, $z\in\mathbb C$ $q(x)$ polynomial. Combining our results bispectral duality obtained in [Mukhin E., Tarasov Varchenko A., Adv. 218 (2008), 216-265, arXiv:math.QA/0605172], relate to $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality trigonometric Gaudin Hamiltonians dynamical acting polynomials $kn$ anticommuting variables.