### Binet Forms Involving Golden Ratio and Two Variables: Convolution Identities

R. Rangarajan, Honnegowoda C.K.

#### Abstract

The irrational number $$\Phi = \frac{1+\sqrt{5}}{2}$$ or $$\phi =\frac{−1+\sqrt{5}}{2}$$ is well known as golden ratio.The binet forms $$L_n = \Phi^n + (−\phi)^n$$ and $$F_n = \frac{\Phi^n−(−\phi)^n}{\sqrt{p5}}$$ define the well known Lucas and Fibonacci numbers. In the present paper, we generalize the binet forms $$\Phi_n(x,y) = \frac{1}{y\cdot \sqrt{5}}[(x + y\Phi)^n−(x−y\phi)^n]$$ and $$\pi_n(x,y)=[(x + y\Phi)^n +(x−y\phi)^n]$$. As a result we obtain a pair of two variable polynomial which are new combinatorial entities. Many convolution identities of Ln and Fn are getting added to the recent literature. A generalized convolution identities will be a worthy enrichment of such combinatorial identities to the current literature.

#### Keywords

Golden ratio; Binet forms; Combinatorial identities

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DOI: http://dx.doi.org/10.26713%2Fjims.v10i1-2.777

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