Higher-order identities for balancing numbers

Abstract

Let $B_n$ be the $n$-th balancing number. In this paper, we give some explicit expressions of $\sum_{l=0}^{2 r-3}(-1)^l\binom{2 r-3}{l}\sum_{j_1+\cdots+j_r=n-2 l\atop j_1,\dots,j_r\ge 1}B_{j_1}\cdots B_{j_r}$ and $\sum_{j_1+\cdots+j_r=n\atop j_1,\dots,j_r\ge 1}B_{j_1}\cdots B_{j_r}$. We also consider the convolution identities with binomial coefficients: $$ \sum_{k_1+\cdots+k_r=n\atop k_1,\dots,k_r\ge 1}\binom{n}{k_1,\dots,k_r}B_{k_1}\cdots B_{k_r} $$ This type can be generalized, so that $B_n$ is a special case of the number $u_n$, where $u_n=a u_{n-1}+b u_{n-2}$ ($n\ge 2$) with $u_0=0$ and $u_1=1$.