I was going through papers such as this one and this one, hoping to find an explicit formula for the $n$-th Fibonacci numbers as an infinite sum. I could not find any. Surely, there are quite a few interesting infinite sums with $F_n$ in the summand, and finite-sum representations of $F_n$, but
I am interested in 'combinatorial' (interpret this qualifier as broadly as you please) functions $f(a,b)$ with $$\sum_{k = 1}^\infty f(n,k) = F_n$$ Where $F_n$ is the $n$-th Fibonacci number.
Obligatory vague catch-all: no trivial solutions!
One way to represent the Fibonacci numbers through infinite sums may be obtained by considering the very interesting family of functions that are called Generalized Binomial Series in the renowned "Concrete Mathematics"(1) treatise.
Please refer thereto for a wide general analysis while here I will summarize without demonstration some properties of interest to our particular aim.
The Generalized Binomial Series is defined as $$ \eqalign{ & {\cal B}_{\,t} (z)\quad \left| {\;t \in \mathbb R} \right.\quad = \sum\limits_{0\, \le \,k} {{{\left( {t\,k} \right)^{\,\underline {\,k - 1\,} } } \over {k!}}z^{\,k} } = \cr & = \sum\limits_{0\, \le \,k} {{1 \over {\left( {tk - k + 1} \right)}}\binom{ t\,k }{ k } z^{\,k} } = 1 + z\sum\limits_{0\, \le \,k} {{1 \over {k + 1}}\binom{ t\,k + t }{ k } z^{\,k} } \cr} $$ and it comes that its powers also have a simple representation $$ \eqalign{ & {\cal B}_{\,t} (z)^{\,r} \quad \left| {\;t,r \in \mathbb R} \right.\quad = \sum\limits_{0\, \le \,k} {{r \over {t\,k + r}}\binom{ t\,k + r }{ k } z^{\,k} } = \cr & = \sum\limits_{0\, \le \,k} {{{r\left( {t\,k + r - 1} \right)^{\,\underline {\,k - 1\,} } } \over {k!}}z^{\,k} } = \cr & = 1 + rz\sum\limits_{0\, \le \,k} {{1\over {k + 1}}\binom{ \,t\,k + t + r - 1 }{ k } z^{\,k} } \cr} $$
Now we have that for $t=1/2, \, r= 1/2$ $$ \eqalign{ & {\cal B}_{\,\,1/2} (z)^{\,\,1/2} = {{z + \sqrt {z^{\,2} + 4} } \over 2} = \exp \left( {{\rm arcsinh}\left( {{z \over 2}} \right)} \right) \cr & - {\cal B}_{\,\,1/2} ( - z)^{\,\,1/2} = - {\cal B}_{\,\,1/2} (z)^{\,\, - \,1/2} = {{z - \sqrt {z^{\,2} + 4} } \over 2} = - \exp \left( { - \,{\rm arcsinh}\left( {{z \over 2}} \right)} \right) \cr} $$
Therefore $$ \eqalign{ & F_{\,n} \quad \left| {\;n \in \,Z\;} \right.\quad = {{\varphi ^{\,\;n} - \hat \varphi ^{\,\;n} } \over {\sqrt 5 }} = \cr & = {1 \over {\sqrt 5 }}\left( {{\cal B}_{\,\,1/2} (1)^{\,\,n/2} - \left( { - 1} \right)^{\,n} {\cal B}_{\,\,1/2} ( - 1)^{\,\,n/2} } \right) = \cr & = {1 \over {\sqrt 5 }}\left( {1 + {n \over 2}\sum\limits_{0\, \le \,k} {{1 \over {k + 1}}\left( \matrix{ \left( {k + n - 1} \right)/2 \cr k \cr} \right)} - \left( { - 1} \right)^{\,n} \left( {1 - {n \over 2}\sum\limits_{0\, \le \,k} {{{\left( { - 1} \right)^k } \over {k + 1}}\left( \matrix{ \left( {k + n - 1} \right)/2 \cr k \cr} \right)} } \right)} \right) = \cr & = {1 \over {\sqrt 5 }}\left( {\left( {1 - \left( { - 1} \right)^{\,n} } \right) + {n \over 2}\sum\limits_{0\, \le \,k} {{{1 + \left( { - 1} \right)^{\,n + k} } \over {k + 1}}\left( \matrix{ \left( {k + n - 1} \right)/2 \cr k \cr} \right)} } \right) = \cr & = {1 \over {\sqrt 5 }}\left( {2\bmod (n,2) + {n \over 2}\sum\limits_{0\, \le \,k} {{{2\left[ {k \equiv n\;\left( {\bmod 2} \right)} \right]} \over {k + 1}}\left( \matrix{ \left( {k + n - 1} \right)/2 \cr k \cr} \right)} } \right) = \cr & = {1 \over {\sqrt 5 }}\left( {2\bmod (n,2) + n\sum\limits_{0\, \le \,j} {{1 \over {2j + \bmod (n,2) + 1}}\left( \matrix{ \left( {2j + \bmod (n,2) + n - 1} \right)/2 \cr 2j + \bmod (n,2) \cr} \right)} } \right) = \cr & = {1 \over {\sqrt 5 }}\left( {2\bmod (n,2) + n\sum\limits_{0\, \le \,j} {{1 \over {2j + \bmod (n,2) + 1}}\left( \matrix{ j + \left\lceil {n/2} \right\rceil - 1/2 \cr 2j + \bmod (n,2) \cr} \right)} } \right) \cr} $$ where the square brackets denote the Iverson bracket.
For instance, for $n=10$ the first partial sums from the expression above give $$F_{10} = 55 : 4.47213595, 22.91969677, 43.9037972, 53.27169917, 54.93059848, 55.00223277, 54.999822, 55.00002003, 54.99999728, 55.00000042, 54.99999993$$
(1)"Concrete Mathematics: a foundation for computer science" R. L. Graham - D.E. Knuth - O. Patshnik - Addison-Wesley 2nd Ed. 1994