Fibonacci numbers and orthogonal polynomials

Abstract

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.

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

References in corpus (1)

• One Parameter Generalizations of the Fibonacci and Lucas Numbers

Cited by in corpus (2)

• The smallest eigenvalue of Hankel matrices
• A common formula for certain generalized Hankel transforms