Higher-order identities for balancing numbers

Takao Komatsu; Prasanta Kumar Ray

paper

Higher-order identities for balancing numbers

arXiv:1608.05925

Abstract

Let $B_{n}$ be the $n$ -th balancing number. In this paper, we give some explicit expressions of $\sum_{l = 0}^{2 r - 3} (- 1)^{l} (l 2 r - 3) \sum_{j _{1} , \dots , j _{r} \geq 1 j _{1} + \dots + j _{r} = n - 2 l} B_{j_{1}} \dots B_{j_{r}}$ and $\sum_{j _{1} , \dots , j _{r} \geq 1 j _{1} + \dots + j _{r} = n} B_{j_{1}} \dots B_{j_{r}}$ . We also consider the convolution identities with binomial coefficients: $k _{1} , \dots , k _{r} \geq 1 k _{1} + \dots + k _{r} = n \sum (k _{1} , \dots , k _{r} n) B_{k_{1}} \dots B_{k_{r}}$ This type can be generalized, so that $B_{n}$ is a special case of the number $u_{n}$ , where $u_{n} = a u_{n - 1} + b u_{n - 2}$ ( $n \geq 2$ ) with $u_{0} = 0$ and $u_{1} = 1$ .