On the m-extension of the Fibonacci and Lucas p-numbers

E. Gokcen Kocer, Naim Tuglu and Alexey Stakhov

Chaos, Solitons & Fractals, 2009, vol. 40, issue 4, 1890-1906

Abstract: In this article, we define the m-extension of the Fibonacci and Lucas p-numbers (p⩾0 is integer and m>0 is real number) from which, specifying p and m, classic Fibonacci and Lucas numbers (p=1, m=1), Pell and Pell–Lucas numbers (p=1, m=2), Fibonacci and Lucas p-numbers (m=1), Fibonacci m-numbers (p=1), Pell and Pell–Lucas p-numbers (m=2) are obtained. Afterwards, we obtain the continuous functions for the m-extension of the Fibonacci and Lucas p-numbers using the generalized Binet formulas. Also we introduce in the article a new class of mathematical constants – the Golden (p,m)-Proportions, which are a wide generalization of the classical golden mean, the golden p-proportions and the golden m-proportions. The article is of fundamental interest for theoretical physics where Fibonacci numbers and the golden mean are used widely.

Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:40:y:2009:i:4:p:1890-1906

DOI: 10.1016/j.chaos.2007.09.071

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