Zeckendorf family identities generalized

Details Zeckendorf family identities generalized Zeckendorf family identities generalized

2011-03-23

arxiv.org/abs/1103.4507v1

Mathematics

Combinatorics

7 pages. This is a brief version. More details can be found at http://www.cip.ifi.lmu.de/~grinberg/zeckendorfLONG.pdf (sourcec

Scientific paper

Philip Matchett Wood and Doron Zeilberger have constructed identities for the Fibonacci numbers f_n of the form 1f_n = f_n for all n >= 1; 2f_n = f_{n-2} + f_{n+1} for all n >= 3; 3f_n = f_{n-2} + f_{n+2} for all n >= 3; 4f_n = f_{n-2} + f_{n} + f_{n+2} for all n >= 3; ...; kf_n = sum of f_{n+s} with s ranging over some finite set S which depends only on k and does not contain any two consecutive integers. These identities are generalized to the case of arbitrary sums of f_{n+a} instead of just multiples of f_n on the left hand sides. The resulting theorem is proved using the approximative properties of the golden ratio.

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