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Generating functions via Hankel and Stieltjes matrices. (English) Zbl 0961.15018
Summary: When the Hankel matrix formed from the sequence \(1,a_1,a_2, \dots\) has an LDLT 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.

MSC:
15B57 Hermitian, skew-Hermitian, and related matrices
15A23 Factorization of matrices
05A15 Exact enumeration problems, generating functions
Software:
OEIS
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