Fibonacci numbers and orthogonal polynomials

TY - JOUR

T1 - Fibonacci numbers and orthogonal polynomials

AU - Berg, Christian

PY - 2011/7

Y1 - 2011/7

N2 - 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.

AB - 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.

KW - Faculty of Science

U2 - 10.1016/j.ajmsc.2011.01.001

DO - 10.1016/j.ajmsc.2011.01.001

M3 - Journal article

SN - 1319-5166

VL - 17

SP - 75

EP - 88

JO - Arab Journal of Mathematical Sciences

JF - Arab Journal of Mathematical Sciences

IS - 2

ER -