Characterizations of \(\Delta\)-Volterra lattice: a symmetric orthogonal polynomials interpretation (Q497745)

Let \(\Delta_t\) be the forward difference operator: \(\Delta_t g (t)=g(t+1)-g(t)\). The main goal of this paper is to obtain characterizations of the \(\Delta\)-Volterra lattice \[ \Delta_t \Gamma (t)= B(t) \Gamma(t)-\Gamma(t+1) B(t), \] where \[ \Gamma (t)= \begin{pmatrix} 0 & 1 & 0 & & & \\ \gamma_1(t) & 0 & 1 & 0 & & \\ 0 & \gamma_2(t) & 0 & 1 & \ddots & \\ & \ddots & \ddots & \ddots & \ddots & \end{pmatrix}, \] \[ B(t)=\begin{pmatrix} \gamma_1(t+1) & 0 & & \\ 0 & \gamma_1(t+1) & 0 & \\ \eta_1(t) & 0 & \gamma_1(t+1) & \ddots \\ 0 & \eta_2(t) & 0 & \ddots \\ & \ddots & \ddots & \ddots \\ \end{pmatrix} \] and \[ \eta_1(t)=\gamma_1(t+1) \gamma_2(t+1), \;\;\eta_n(t)=\frac{\gamma_1(t+1)\cdots \gamma_{n+1}(t+1)}{\gamma_1(t)\cdots \gamma_{n-1}(t)}\, , \;\;n=2, \dots, \] with \(1+\gamma_1(t+1)\neq 0\) and \(\gamma_{n}(t)\neq 0\). In Section 2, the authors present the main theorems of the \(\Delta\)-Volterra lattices: a representation of the symmetric orthogonality functional and a Lax-type theorem. In Section 3, it is shown that the solutions of \(\Delta\)-Toda lattice are connected to \(\Delta\)-Volterra lattice through Miura or Bäcklund transformations. Finally, in Section 4, an explicit example of solutions of \(\Delta\)-Volterra and \(\Delta\)-Toda lattices related to Jacobi polynomials is given, and connected with the results presented in this paper.