Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel

Let $M$ be the tensor product of finite-dimensional polynomial evaluation $Y(\mathfrak{gl}_N)$-modules. Consider the universal difference operator $\mathfrak D=\sum\limits_{k=0}^N (-1)^k\mathfrak T_k(u) e^{-k\partial _u }$ whose coefficients $\mathfrak T_k(u)\colon M\to M$ are the XXX transfer matrices associated with $M$. We show that the difference equation $\mathfrak D f=0$ for an $M$-valued function $f$ has a basis of solutions consisting of quasi-exponentials. We prove the same for the universal differential operator $D=\sum\limits_{k=0}^N (-1)^k\mathcal S_k(u)\partial_u^{N-k}$ whose coefficients $\mathcal S_k(u)\colon\mathcal M\to\mathcal M$ are the Gaudin transfer matrices associated with the tensor product $\mathcal M$ of finite-dimensional polynomial evaluation $\mathfrak{gl}_N[x]$-modules.