Bispectral and (gl_N, gl_M) Dualities, Discrete Versus Differential

Let $V = < x^{\lambda_i}p_{ij}(x), i=1,...,n, j=1, ..., N_i >$ be a space of quasi-polynomials in $x$ of dimension $N=N_1+...+N_n$. The regularized fundamental differential operator of $V$ is the polynomial differential operator $\sum_{i=0}^N A_{N-i}(x)(x \frac d {dx})^i$ annihilating $V$ and such that its leading coefficient $A_0$ is a monic polynomial of the minimal possible degree. Let $U = < z_a^{u} q_{ab}(u), a=1,...,m, b=1,..., M_a >$ be a space of quasi-exponentials in $u$ of dimension $M=M_1+...+M_m$. The regularized fundamental difference operator of $U$ is the polynomial difference operator $\sum_{i=0}^M B_{M-i}(u)(\tau_u)^i$ annihilating $U$ and such that its leading coefficient $B_0$ is a monic polynomial of the minimal possible degree. Here $(\tau_uf)(u)=f(u+1)$. Having a space $V$ of quasi-polynomials with the regularized fundamental differential operator $D$, we construct a space of quasi-exponentials $U = <z_a^{u}q_{ab}(u) >$ whose regularized fundamental difference operator is the difference operator $\sum_{i=0}^N u^i A_{N-i}(\tau_u)$. The space $U$ is constructed from $V$ by a suitable integral transform. Similarly, having $U$ we can recover $V$ by a suitable integral transform. Our integral transforms are analogs of the bispectral involution on the space of rational solutions to the KP hierarchy \cite{W}. As a corollary of the properties of the integral transforms we obtain a correspondence between solutions to the Bethe ansatz equations of two $(gl_N, gl_M)$ dual quantum integrable models: one is the special trigonometric Gaudin model and the other is the special XXX model.