Journal of Integer Sequences, Vol. 24 (2021), Article 21.5.1

Generalized Catalan Recurrences, Riordan Arrays, Elliptic Curves, and Orthogonal Polynomials

Paul Barry
School of Science
Waterford Institute of Technology


We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos-4 sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos-4 sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions.

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(Concerned with sequences A000045 A000108 A001003 A001519 A004148 A004149 A006318 A006769 A014431 A023431 A025235 A025243 A025272 A048990 A051255 A084782 A085139 A086581 A091561 A128720 A157101 A162985 A174171 A174403 A174404 A178376 A178384 A178622 A178627 A187256 A236337.)

Received March 28 2019; revised version received April 19 2021. Published in Journal of Integer Sequences, April 19 2021.

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