On biorthogonal polynomials and deformed Volterra hierarchies

In this paper, we investigate a family of biorthogonal polynomials arising from the Ising model coupled to two-dimensional quantum gravity. In contrast to the standard threeterm recurrence relations characteristic of ordinary orthogonal polynomials, these biorthogonal polynomials satisfy a recurrence relation with a gap of 2. This structural peculiarity naturally gives rise to an integrable hierarchy that generalizes the classical Volterra lattice. Continuous and discrete evolutions are both considered.