Generating Functions via Hankel and Stieltjes Matrices

When the Hankel matrix formed from the sequence 1, a1, a2, ... has an LDL T decomposition, we provide a constructive proof that the Stieltjes matrix SL 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 SL must have, and we demonstrate, constructively, a oneto-one correspondence between the generating function for the sequence and SL. If L is Riordan when using ordinary generating functions, we show how to derive a recurrence relation for the sequence.