Toda chain, Stieltjes function, and orthogonal polynomials

We discuss relations between the theory of orthogonal polynomials, Hankel determinants, and the unrestricted one-dimensional Toda chain. In particular, we show that the equations of motion for the Toda chain are equivalent to a Riccati equation for the Stieltjes function. We consider some examples of the Stieltjes function with an explicit (hypergeometric and elliptic) time dependence in detail.