Some Binomial Fibonacci Identities

Calvin T. Long

A chapter in Applications of Fibonacci Numbers, 1990, pp 241-254 from Springer

Abstract: Abstract Interest in binomial Fibonacci identities goes back at least to E. Lucas [8] who obtained formulas like (1) $$\sum\limits_{i = 0}^r {\left( {_i^r} \right)} {F_{n + 1 = }}{F_{n + 2r}} and \sum\limits_{i = 0}^r {\left( {_i^r} \right){L_{n + i}}} = {L_{n + 2r}}$$ 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 $${F^n}{\left( {F + 1} \right)^r} = {F^n}{\left( {{F^2}} \right)^r} = {F^{n + 2r}}$$ (1′) $${L^n}{\left( {L + 1} \right)^r} = {L^n}{\left( {{L^2}} \right)^r} = {L^{n + 2r}}$$ 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].

Date: 1990
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DOI: 10.1007/978-94-009-1910-5_28

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