Incomplete balancing and lucas-balancing numbers

The terms balancing numbers and Lucas-balancing numbers are used to describe the series of numbers generated by the recursive formulas Bn = 6Bn−1 − Bn−2; B0 = 0, B1 = 1 with n ≥ 2 and Cn = 6Cn−1 − Cn−2; C0 = 1, C1 = 3, with n ≥ 2 respectively [1,10]. The roots λ1 = 3+ √ 8 and λ2 = 3− √ 8 for both these sequences, the respective Binet formulas are Bn = λ1−λ2 λ1−λ2 and Cn = λ1+λ2 2 [1, 10]. Many interesting results of balancing numbers and their related sequences can be found in [5, 10–12]. In [4], Filipponi established two interesting classes of integers namely, incomplete Fibonacci numbers and incomplete Lucas numbers which were obtained from some of the well-known combinatorial forms of Fibonacci and Lucas numbers. He has also studied some of the congruence properties for incomplete Lucas numbers in [4]. Filipponi dreamt a glimpse of possible generalizations of incomplete Fibonacci and Lucas numbers which were fulfilled by some authors later [2, 7–9, 13]. In this article we establish some combinatorial expressions for balancing and Lucas-balancing numbers and introduce incomplete balancing and incomplete Lucas-balancing numbers. Balancing sequence has been generalized in many ways. One important generalization of balancing numbers is the k-balancing numbers introduced in [6, 12]. k-balancing numbers are defined recursively by