Generating Functions via Hankel and Stieltjes Matrices

When the Hankel matrix formed from the sequence 1, a_1, a_2, ... has an L*D*L^T decomposition, we provide a constructive proof that the Stieltjes matrix S_L associated with L is tridiagonal. In the important case when L is a Riordan matrix using ordinary or exponential generating functions, we determine the specific form that S_L must have, and we demonstrate, constructively, a one-to-one correspondence between the generating function for the sequence and S_L. If L is Riordan when using ordinary generating functions, we show how to derive a recurrence relation for the sequence.