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Catalan numbers, the Hankel transform, and Fibonacci numbers. (English) Zbl 1041.11014

The authors prove that the Hankel transformation of a sequence whose \(n\)th element is the sum of the \(n\)th and \((n+1)\)th Catalan numbers is a subsequence of the Fibonacci numbers. They do this by finding the explicit form for the coefficients in the tree-term recurrence relation that the corresponding orthogonal polynomials satisfy.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
05A10 Factorials, binomial coefficients, combinatorial functions
44A15 Special integral transforms (Legendre, Hilbert, etc.)

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