Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.2

Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays

Paul Barry
School of Science
Waterford Institute of Technology

Aoife Hennessy
Department of Computing, Mathematics and Physics
Waterford Institute of Technology


We study constant coefficient four term recurrences for polynomials, in analogy to the three-term recurrences that are associated with orthogonal polynomials. We show that for a family of polynomials obeying such a four-term recurrence, the coefficient array is an ordinary Riordan array of a special type, and vice versa. In certain cases, it is possible to transform these polynomials into related orthogonal polynomials. We characterize the form of the production matrices of the inverse coefficient arrays.

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(Concerned with sequences A000108 A001405 A005043 A007297 A007318 A033184 A064641 A089942 A129147.)

Received November 14 2011; revised version received March 26 2012. Published in Journal of Integer Sequences, March 26 2012.

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