Degree $5$ Fibonacci Sums via the Gelin-Ces\`aro Identity

Abstract

Let $F_k$ be the $k$th Fibonacci number. Let $(G_k)_{k\in\mathbb Z}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We employ the Gerin-Ces\`aro identity and an identity of Brousseau to evaluate the following sums: $\sum_{j=1}^n{(\pm 1)^{j - 1}G_j^5}$, $\sum_{j = 1}^n {G_{j - 1} G_{j} G_{j + 1} G_{j + 2} G_{j + m} }$, $\sum_{j = 1}^n {(-F_{m - 3})^{n - j} ( F_{m + 2} )^j G_{j - 1} G_{j} G_{j + 1} G_{j + 2} G_{j + m} }$, and $\sum_{j = 1}^n {(-F_{m + 2})^{n - 2}F_{m - 3}^jG_{j + m} \left( {G_{j - 2} G_{j - 1} G_j G_{j + 1} G_{j + 2} G_{j + 3} } \right)^{ - 1} }$. Among other results, we evaluate the sum and alternating sum of products of five consectutive Fibonacci-like numbers, namely $\sum_{j = 1}^n {\left( { \pm 1} \right)^{j - 1} G_j G_{j + 1} G_{j + 2} G_{j + 3} G_{j + 4} }$.