Generalized Catalan Recurrences, Riordan Arrays, Elliptic Curves, and Orthogonal Polynomials
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.
Full version: pdf,
(Concerned with sequences
Received March 28 2019; revised version received April 19 2021.
Published in Journal of Integer Sequences,
April 19 2021.
Journal of Integer Sequences home page