Fibonacci numbers and orthogonal polynomials

Details Fibonacci numbers and orthogonal polynomials Fibonacci numbers and orthogonal polynomials

2006-09-11

arxiv.org/abs/math/0609283v2

Mathematics

Number Theory

A note dated June 2007 has been added with some historical comments. Some references have been added and completed

Scientific paper

We prove that the sequence $(1/F_{n+2})_{n\ge 0}$ of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little $q$-Jacobi polynomials with $q=(1-\sqrt{5})/(1+\sqrt{5})$. We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices $(1/F_{i+j+2})$ have integer entries. We prove analogous results for the Hilbert matrices.

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