Summary: When the Hankel matrix formed from the sequence 1, $a_{1}, a_{2}, \dots $has an $LDL^{T}$ 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.

Hankel matrix, Stieltjes matrix, ordinary generating function, exponential generating function, riordan matrix, LDU decomposition, tridiagonal matrix