q-discretization of the Kostant–Toda equation and its asymptotic analysis

The famous Toda equation is an integrable system related to similarity transformations of tridiagonal matrices. The discrete Toda equation, which is a time-discretization of the Toda equation, is essentially the recursion formula of the quotient-difference (qd) algorithm for computing eigenvalues of tridiagonal matrices. Another time-discretization of the Toda equation is the q-discrete Toda equation, which is derived by replacing standard derivatives with the so-called q-derivatives that involves a parameter q such that $0<q<1$. In a prior work, we related the q-discrete Toda equation to implicit-shift $LR$ transformations (which are similarity transformations) of tridiagonal matrices. Furthermore, we developed the determinantal solution to clarify the convergence as discrete-time goes to infinity. In this paper, we propose an extension of the q-discrete Toda equation as a time-discretization of the Kostant–Toda equation and then show the convergence as discrete-time goes to infinity from the perspective of implicit-shift $LR$ transformations of Hessenberg matrices. We also present numerical examples to verify the convergence as discrete-time goes to infinity in the proposed q-discrete Kostant–Toda equation.