Fibonacci numbers and orthogonal polynomials

We prove that the sequence (1/Fn+2)n⩾0 of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability measure and we identify the orthogonal polynomials as little q-Jacobi polynomials with q=1-5/1+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/Fi+j+2) have integer entries. We prove analogous results for the Hilbert matrices.