Generalized stirling numbers, Exponential riordan arrays, and toda chain equations

We study the properties of three families of exponential Riordan arrays related to the Stirling numbers of the first and second kind. We relate these exponential Riordan arrays to the coefficients of families of orthogonal polynomials. We calculate the Hankel transforms of the moments of these orthogonal polynomials. We show that the Jacobi coefficients of two of the matrices studied satisfy generalized Toda chain equations. We finish by defining and characterizing the elements of an exponential Riordan array associated to generalized Stirling numbers studied by Lang.