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: Gn=aGn−1+Gn−2 (n is even) or Gn=bGn−1+Gn−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 ∑k=n∞(a/b)ξ(k+1)GkGk+m−1,m=0,2,4,⋯, and ∑k=n∞1GkGk+m−1,m=1,3,5,⋯.