A combinatorial proof of a family of multinomial-Fibonacci identities (Q2926274)

In the paper, there are proved via both combinatorial and algebraic methods the following two theorems.NEWLINENEWLINE NEWLINETheorem 1. For \(n \geq 0\), \(r \geq 1\), \(a_i \geq 0\) for \(0 \leq i \leq r\) and \(x_i \geq 0\) for \(1 \leq i \leq r\), NEWLINE\[NEWLINE\sum_{\sum\limits_{i=0}^r a_i = n} \left ( \begin{matrix} n \\ a_0, ..., a_r \end{matrix} \right ) \left (\sum_{i=0}^r F_{x_i - 2} \right )^{a_0} F_{\sum\limits_{i=0}^r a_i x_i} = \left (\sum_{i=1}^r F_{x_i} \right )^n F_{2n},NEWLINE\]NEWLINE where \(\sum\limits_{i=1}^r F_{x_i} \geq 1\).NEWLINENEWLINETheorem 2. For \(n \geq 0\), \(r \geq 1\), \(a_i \geq 0\) for \(0 \leq i \leq r\) and \(x_i \geq 0\) for \(1 \leq i \leq r\), NEWLINE\[NEWLINE\sum_{\sum\limits_{i=0}^r a_i = n} \left ( \begin{matrix} n \\ a_0, ..., a_r \end{matrix} \right ) \left (\sum_{i=0}^r p_i F_{x_i - 2} \right )^{a_0}\left (\prod_{i=1}^r p_i^{a_i} \right ) F_{\sum\limits_{i=0}^r a_i x_i} = \left(\sum_{i=1}^r p_i F_{x_i} \right )^n F_{2n},NEWLINE\]NEWLINE where \(\sum\limits_{i=1}^r p_i F_{x_i} \geq 1\).