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Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials
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abstract
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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Elliptic Curves, Riordan arrays and Lattice Paths
Elliptic curves of a specific form correspond to lattice paths whose Hankel transform reproduces the curve's elliptic divisibility sequence.
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