In this paper, we study the reciprocal sums of products of Fibonacci and Lucas numbers. Some identities are obtained related to the numbers ??,k=n 1/FkLk+m and ??,k=n 1/LkFk+m, m ? 0.

In this paper we consider a generalized bi-periodic Fibonacci $\left\{f_n\right\}$ and a generalized bi-periodic Lucas sequence $\left\{q_n\right\}$ which are respectively defined by $f_0=0$, $f_1=1$, $f_n=a{f}_{n-1}+c{f}_{n-2}$ (n is even) or $f_n=b{f}_{n-1}+c{f}_{n-2}$ (n is odd), and $q_0=2d$, $q_1=ad$, $q_n=b{q}_{n-1}+c{q}_{n-2}$ (n is even) or $q_n=a{f}_{n-1}+c{q}_{n-2}$ (n is odd). We obtain various relations between these two sequences.

Recently Panda et al. obtained some identities for the reciprocal sums of balancing and Lucas-balancing numbers. In this paper, we derive general identities related to reciprocal sums of products of two balancing numbers, products of two Lucas-balancing numbers and products of balancing and Lucas-balancing numbers. The method of this paper can also be applied to even-indexed and odd-indexed Fibonacci, Lucas, Pell and Pell–Lucas numbers.

This paper concerns the properties of the generalized bi-periodic Fibonacci numbers $\left\{G_n\right\}$ generated from the recurrence relation: $G_n=a{G}_{n-1}+G_{n-2}$ (n is even) or $G_n=b{G}_{n-1}+G_{n-2}$ (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 ${\left({\sum }_{k=n}^{\infty }\frac{{(a/b)}^{\xi (k+1)}}{G_k{G}_{k+m}}\right)}^{-1},m=0,2,4,\cdots ,$ and ${\left({\sum }_{k=n}^{\infty }\frac{1}{G_k{G}_{k+m}}\right)}^{-1},m=1,3,5,\cdots .$