Geodesic
Generating functions via Hankel and Stieltjes matrices
Journal of integer sequences, Tome 3 (2000) no. 2
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.
Keywords: Hankel matrix, Stieltjes matrix, ordinary generating function, exponential generating function, riordan matrix, LDU decomposition, tridiagonal matrix 
@article{JIS_2000__3_2_a5,
     author = {Peart,  Paul and Woan,  Wen-Jin},
     title = {Generating functions via {Hankel} and {Stieltjes} matrices},
     journal = {Journal of integer sequences},
     year = {2000},
     volume = {3},
     number = {2},
     zbl = {0961.15018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2000__3_2_a5/}
}
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Peart,  Paul; Woan,  Wen-Jin. Generating functions via Hankel and Stieltjes matrices. Journal of integer sequences, Tome 3 (2000) no. 2. http://geodesic.mathdoc.fr/item/JIS_2000__3_2_a5/