On the Reciprocal Sums of Products of Two Generalized Bi-Periodic Fibonacci Numbers

Abstract

This paper concerns the properties of the generalized bi-periodic Fibonacci numbers {Gn} generated from the recurrence relation: G(n) = aG(n-1)+G(n-2) (n is even) or G(n) = bG(n-1) + G(n-2) (n is odd). We derive general identities for the reciprocal sums of products of two generalized bi-periodic Fibonacci numbers. More precisely, we obtain formulas for the integer parts of the numbers (Sigma(infinity)(k=n) (a/b)(xi(k+1)/)G(k)G(k+m))(-1) ,m = 0,2,4 ..., and (Sigma(infinity)(k=n)1/G(k)G(k+m))(-1), m =1,3,5, ....