On several kinds of sums of balancing numbers

The balancing numbers B-n (n = 0,1, ...) are solutions of the binary recurrence B-n = 6B(n-1) - Bn-2 (n >= 2) with B-0 = 0 and B-1 = 1. In this paper we show several relations about the sums of product of two balancing numbers of the type Sigma(m=0Bkm+r Bk(n-m)+r)-B-n (k > r >= 0) and the alternating sum of reciprocal of balancing numbers [Sigma(infinity )(k=n)1/B-lk)(-1)]. Similar results are also obtained for Lucas-balancing numbers C-n (n 0, 1, ...), satisfying the binary recurrence C-n = 6C(n-1) - Cn-2 (n >= 2 ) with C-0 = 1 and C-1 = 3. Some binomial sums involving these numbers are also explored.