Binet Forms Involving Golden Ratio and Two Variables: Convolution Identities (Journal of Informatics and Mathematical Sciences, 2018)
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