Abstract
A well-known identity is
where F k and L k are the Fibonacci and Lucas numbers, respectively. With the definitions \(F_{-k} = (-1)^{k + 1}F_k\) and \(L_{-k} = (-1)^kL_k\) formula (1.1) is true for all integer m and n. The identity is easy to prove, and it is evidently useful; Rokach [11], for example, proved that (1.1) is about 2.88 times more efficient than the usual Fibonacci recurrence for computing the Fibonacci numbers.
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References
Bruckman, P.S. “Solution to Problem H-487 (Proposed by S. Rabinowitz).” The Fibonacci Quarterly, Vol. 33 (1995): p. 382.
Comtet, L. Advanced Combinatorics. Dordrecht: Reidel, 1974.
Filipponi, P. and Horadam, A.F. “Derivative Sequences of Fibonacci and Lucas Polynomials.” Applications of Fibonacci Numbers. Volume 4. Edited by G.E. Bergum, A.F. Horadam and A.N. Philippou. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991: pp. 99–108.
Filipponi, P. and Horadam, A.F. “Integration Sequences of Fibonacci and Lucas Polynomials.” Applications of Fibonacci Numbers. Volume 5. Edited by G.E. Bergum, A.F. Horadam and A.N. Philippou. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993: pp. 317–330.
Horadam, A.F. “Generating Functions for Powers of a Certain Generalized Sequence of Numbers.” Duke Math. Journal, Vol. 32 (1965): pp. 437–446.
Horadam, A.F. “Jacobsthal Representation Polynomials.” The Fibonacci Quarterly, Vol. 35 (1997): pp. 137–148.
Horadam, A.F. “Basic Properties of a Certain Generalized Sequence of Numbers.” The Fibonacci Quarterly, Vol. 3 (1965): pp. 161–176.
Horadam, A.F. and Mahon, J.M. “Pell and Pell-Lucas Polynomials.” The Fibonacci Quarterly, Vol. 23 (1985): pp. 7–20.
Howard, F.T. “Lacunary Recurrences for Sums of Powers of Integers.” The Fibonacci Quarterly, Vol. 36 (1998): pp. 435–442.
Rabinowitz, S. “Algorithmic Manipulation of Second Order Linear Recurrences.” The Fibonacci Quarterly (to appear).
Rakoch, A. “Optimal Computations, by Computer, of Fibonacci Numbers.” The Fibonacci Quarterly, Vol. 34 (1996): pp. 436–439.
Vajda, S. Fibonacci and Lucas Numbers, and the Golden Section. Ellis Horwood Limited, Chichester, 1989.
Waddill, M.E. and Sacks, L. “Another Generalized Fibonacci Sequence.” The Fibonacci Quarterly, Vol. 5 (1967): pp. 209–227.
Wilf, H.S. Generatingfunctionology. Academic Press, Inc., San Diego, 1990.
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Howard, F.T. (1999). Generalizations of a Fibonacci Identity. In: Howard, F.T. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4271-7_20
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DOI: https://doi.org/10.1007/978-94-011-4271-7_20
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