Some Binomial Fibonacci Identities

Abstract

Interest in binomial Fibonacci identities goes back at least to E. Lucas [8] who obtained formulas like

$$\sum\limits_{i = 0}^r {\left( {_i^r} \right)} {F_{n + 1 = }}{F_{n + 2r}} and \sum\limits_{i = 0}^r {\left( {_i^r} \right){L_{n + i}}} = {L_{n + 2r}}$$

((1))

where F _n and L _n represent, respectively, the nth Fibonacci and Lucas numbers. Indeed, Lucas used the Binet formulas and the characteristic equation x² = x + 1 to argue that equations (1) can be written in the form

$${F^n}{\left( {F + 1} \right)^r} = {F^n}{\left( {{F^2}} \right)^r} = {F^{n + 2r}}$$

$${L^n}{\left( {L + 1} \right)^r} = {L^n}{\left( {{L^2}} \right)^r} = {L^{n + 2r}}$$

((1′))

where, after simplification, the powers of F and L are replaced by the appropriately subscripted F’s and L’s. This same approach for other identities was investigated further by Hoggatt and Lind in [4] and Ruggles [9].