Degree $5$ Fibonacci Sums via the Gelin-Cesàro Identity

Let $F_{k}$ be the $k$ th Fibonacci number. Let $(G_{k})_{k \in Z}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We employ the Gerin-Cesàro 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 $\sum_{j = 1}^{n} (- F_{m + 2})^{n - 2} F_{m - 3}^{j} G_{j + m} {(G_{j - 2} G_{j - 1} G_{j} G_{j + 1} G_{j + 2} G_{j + 3})}^{- 1}$ . Among other results, we evaluate the sum and alternating sum of products of five consectutive Fibonacci-like numbers, namely $\sum_{j = 1}^{n} {(\pm 1)}^{j - 1} G_{j} G_{j + 1} G_{j + 2} G_{j + 3} G_{j + 4}$ .

Degree $5$ Fibonacci Sums via the Gelin-Cesàro Identity

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Degree $5$ Fibonacci Sums via the Gelin-Cesàro Identity

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Degree 55 Fibonacci Sums via the Gelin-Cesàro Identity

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Degree 55 Fibonacci Sums via the Gelin-Cesàro Identity

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Degree $5$ Fibonacci Sums via the Gelin-Cesàro Identity

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Degree $5$ Fibonacci Sums via the Gelin-Cesàro Identity

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