Some Reciprocal Summation Identities with Applications to the Fibonacci and Lucas Numbers

Abstract

With the use of theta functions and some series rearrangement we present a summation identity, involving the Fibonacci and Lucas numbers, of the form $$ \sum {{a_n}{b_n} = \sum {{a_n}\sum {{b_n}} } } $$ (1) for sequences {a n} and {b n}. It may be of some interest to determine other such examples. Two other reciprocal summation identities for the Fibonacci and Lucas numbers are derived from the proof of our first result. We also present a short elementary proof of a sum involving Lucas numbers. This sum is of interest since it is rational, whereas related sums such as $$ \sum\limits_{n = 1}^\infty {\frac{1}{{{F_2}n}}} {\rm{ and }}\sum\limits_{n = 1}^\infty {\frac{1}{{{L_2}n}}} $$ (2) have been shown to be irrational (see [1], [2] and [3]). As usual in the following theorems F n and L n denote the n th Fibonacci and Lucas number, respectively.