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

Full Text:

PDF

References


W.S. Anglin, The Queen of Mathematics, An Introduction to Number Theory, Kluwer Academy Publishers (1995).

T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley Interscience Publication, New York (2001).

T. Koshy, Elementary Number Theory with Applications, second edition, Academic Press (2007).

A. Kim, Convolution sums related to fibonacci numbers and lucas numbers, Asian Research Journal of Mathematics 1(1) (2016), 1 – 17.

A. Kim, Generalization of convolution sums with fibonacci numbers and lucas numbers, Asian Research Journal of Mathematics 1(1) (2016), 1 – 10.

J.C. Mason and D.C. Handscomb, Chebyshev Polynomials, CRC Press LLC, New York (2003).

R. Rangarajan, Honnegowda C.K. and Shashikala P., Certain combinatorial results on two variable hybrid Fibonacci polynomials, International Journal of Computational and Applied Mathematics 12(2) (2017), 603 – 613.

R. Rangarajan, Honnegowda C.K. and Shashikala P., A two variable Lucas polynomials corresponding to hybrid Fibonacci polynomials, Global Journal of Pure and Applied Mathematics 12(6) (2017), 1669 – 1682.

R. Rangarajan, Honnegowda C.K. and Rangaswamy, Convolution identities involving fixed power of expanding variable, hybrid Fibonacci and Lucas polynomilas, International J. of Math. Sci. & Engg. Appls. 11(II) (2017), 99 – 110.

R. Rangarajan, Honnegowda C.K. and Rangaswamy, Binomial convolutions identities of hybrid Fibonacci and Lucas polynomials, International J. of Math. Sci. & Engg. Appls. 11(III) (2017), 101 –111.

J. Riordan, Combinatorial Identities, Robert E. Krieger Publishing Company, New York (1979).

J. Riordan, An introduction to Combinatorial Analysis, John Wiley and Sons, Inc., New York (1967).

N. Robbins, Beginning Number Theory, W.M.C. Brown Pub. (1993).

S. Vajda, Fibonacci and Lucas Numbers and Golden Section, Theory and Applications, Ellis-Horwood, London (1989).




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

eISSN 0975-5748; pISSN 0974-875X