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