where $\phi = \frac{1+\sqrt{5}}{2}$ and $k!_F$ is the fibonorial of $k$, or the product of the first $k$ Fibonacci numbers? My hunch is that, this can be represented as a function in terms of the constant in the case where x=1, so I tried the following: $$ \begin{split} S = \sum_{k=0}^{\infty}{\frac{\phi^k}{k!_F}} & =\sum_{k=0}^{\infty}{\frac{F_k\phi+F_{k-1}}{k!_F}} \\ &=\sum_{k=0}^{\infty}\Bigg (\phi\frac{F_{k}}{k!_F}+\frac{F_{k-1}}{k!_F}\Bigg)\\ & =\phi \displaystyle \sum_{k=0}^{\infty}\frac{F_{k}}{k!_F}+\sum_{k=0}^{\infty}\frac{F_{k-1}}{k!_F} \end{split} $$ How should I continue from here?. I'm asking for a closed form expression but any alternate representations as an integral or another series are greatly appreciated as well.

(Fibonacci numbers as defined by the recurence $ F_k = F_{k-1}+ F_{k-2} $ and $ F_0=F_1=1 $)

(Edit: The first couple of terms of $ \displaystyle \sum_{k=0}^{\infty}\frac{1}{(k-2)!_F}\frac{1}{F_k} $ and the first term of $ \displaystyle \sum_{k=0}^{\infty}\frac{1}{(k-1)!_F} $ aren't defined so I changed it back to $ \displaystyle \sum_{k=0}^{\infty}\frac{F_{k-1}}{k!_F} $ and $ \displaystyle \sum_{k=0}^{\infty}\frac{F_{k}}{k!_F}$ respectively )

(Edit: It seems that $\displaystyle \sum_{k=0}^{\infty} \frac{F_k}{k!_F} = \sum_{k=0}^{\infty} \frac{1}{k!_F}$)

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