Generating functions via Hankel and Stieltjes matrices. (English) Zbl 0961.15018

Peart, Paul; Woan, Wen-Jin

J. Integer Seq. 3, No. 2, Art. 00.2.1, 13 p. (2000).

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

Keywords:

factorization; Hankel matrix; LDLT decomposition; Stieltjes matrix; Riordan matrix; generating functions; recurrence relation

Software:

OEIS