On Product Difference Fibonacci Identities
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R. S. Melham
Abstract
Simson's identity is a well-known Fibonacci identity in which the difference of certain order 2 products has a particularly pleasing form. Other old and beautiful identities of a similar nature are attributed to Catalan, Gelin and Cesàro, and Tagiuri. Catalan's identity can be described as a family of product difference Fibonacci identities of order 2 with 1 parameter. In Section 2 of this paper we present four families of product difference Fibonacci identities that involve higher order products. Being self-dual, each of these families may be regarded as a higher order analogue of Catalan's identity. We also state two conjectures that give the form of similar families of arbitrary order. In the final section we give other interesting product difference Fibonacci identities.
© de Gruyter 2011
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- Maximum GCD Among Pairs of Random Integers
- First Remark on a ζ-Analogue of the Stirling Numbers
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- Values of the Euler and Carmichael Functions which are Sums of Three Squares
- The Intrinsic Periodic Behaviour of Sequences Related to a Rational Integral
- Note on the Diophantine Equation Xt + Yt = BZt
- Coefficients in Powers of the Log Series
- On a Combinatorial Conjecture
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Articles in the same Issue
- Maximum GCD Among Pairs of Random Integers
- First Remark on a ζ-Analogue of the Stirling Numbers
- On Product Difference Fibonacci Identities
- New Sequences that Converge to a Generalization of Euler's Constant
- On the Number of Factorizations of an Integer
- Values of the Euler and Carmichael Functions which are Sums of Three Squares
- The Intrinsic Periodic Behaviour of Sequences Related to a Rational Integral
- Note on the Diophantine Equation Xt + Yt = BZt
- Coefficients in Powers of the Log Series
- On a Combinatorial Conjecture
- The (Exponential) Bipartitional Polynomials and Polynomial Sequences of Trinomial Type: Part I
- A Cauchy–Davenport Type Result for Arbitrary Regular Graphs
