A Study on Generalized Balancing Numbers

In this paper, we investigate properties of the generalized balancing sequence and we deal with, in detail, namely, balancing, modified Lucas-balancing and Lucas-balancing sequences. We present Binet’s formulas, generating functions and Simson formulas for these sequences. We also present sum formulas of these sequences. We provide the proofs to indicate how the sum formulas, in general, were discovered. Of course, all the listed sum formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we consider generalized balancing sequence at negative indices and construct the relationship between the sequence and itself at positive indices. This illustrates the recurrence property of the sequence at the negative index. Meanwhile, this connection holds for all integers. Furthermore, we give some identities and matrices related with these sequences.