I obtained the very strange formula above (in the question title) and I'm *starting to understand* what does it mean. It's very similar to Poisson summation and indeed it is possible to get Poisson summation from that form. Natural question seems to be to ask if the equation above is stronger than Poisson summation or is it equivalent (Is it possible to prove my form from Poisson summation) <br> 
* First of all, I point out that the above formula cannot be used for numerical calculations because of its divergence.
* If I understand it correctly, to sum a series from  $-\infty $ to $\infty $ means to broke it in two parts at some point $a$ and to treat each of them as an analytic continuation: if my understanding is not correct and/or you know anything about it or what does it mean, please let me know.

**Proof:** Let $f(x)$ be some an analytic function with a given Taylor series. By using the theorem on divergent summation derived by Ramanujan, we can associate to it the convergent integral $\int_x^{\infty}f (t) dt $ so we can write the summation in terms of Euler–Maclaurin formula as follows 
$$
\sum_{k=x}^{\infty}f (k)=- \sum_{n=0}^{\infty} \frac {(-1)^nf^{(n-1)}(x)B_n}{n!}.
$$
We know also that 
$$
(-1)^nB_n=2 \pi \int_{-\infty}^{\infty} \frac {(-\frac {1}{2}+it)  ^n}{(e^{\pi t}+e^{-\pi t})^2}dt, 
$$ 
so we get 
$$
\begin{split}
\sum_{k=x}^{\infty} f (k) &=-2 \pi \sum_{n=0}^{\infty}\frac{f^{(n-1)} (x)}{n!}\int_{-\infty}^{\infty} \frac {(-\frac {1}{2}+it)  ^n}{(e^{\pi t}+e^{-\pi t})^2}dt\\ 
& \text{ and by using Taylor series }\\
&=-2\pi\int_{-\infty}^{\infty} \frac {F(x-\frac {1}{2}+it)}{(e^{\pi t}+e^{-\pi t})^2}dt.
\end{split}$$

Let's use the consequences of Ramanujan's theorem another time
$$
\begin{split} 
\sum_{k=x}^{\infty} f (k) & = -2\pi\int_{-\infty}^{\infty} \frac {F(x-\frac {1}{2}+it)}{(e^{\pi t}+e^{-\pi t})^2}dt\\
& = 2\pi \int_{\infty}^{\infty} \frac {\int_{0}^{\infty}f(u)du-\int_{0}^{x-\frac {1}{2}+it}\sum_{k=0}^{\infty}\frac{f^{(k)}(0)(u)^k}{k!}du}{(e^{\pi t}+e^{-\pi t})^2}dt\\
& = 2\pi \int_{-\infty}^{\infty} \frac {\int_{0}^{\infty}f(u)du-\sum_{k=0}^{\infty}\frac{f^{(k)}(0)(x-\frac {1}{2}+it)^{k+1}}{k!(k+1)}du}{(e^{\pi t}+e^{-\pi t})^2}dt\\
& = \int_{0}^{\infty}f(u)du+\sum_{k=0}^{\infty}\frac{f^{(k)}(0)\zeta (-k,x)}{k!}.
\end{split}
$$ 
Now let's assume $x=1$: then we can use Riemann's zeta functional equation to derived some transformation of the above expression:
$$
\begin{split} 
\sum_{k=1}^{\infty} f (k) & = \int_{0}^{\infty}f(t)dt+ \sum_{k=0}^{\infty}\frac{f^{(k)}(0)(2 \pi)^{-k-1} (i^{-k-1}+(-i)^{-k-1}) \Gamma (1+k)\zeta (1+k) }{k!} \\ 
& =  \int_{0}^{\infty}f(t)dt+\sum_{n=1}^{\infty}\sum_{k=0}^{\infty} \frac { f^{(k)}(0)(2 \pi)^{-k-1} (i^{-k-1}+(-i)^{-k-1}) n^{-k-1}\Gamma (k+1)}{k!} \\ 
& = \int_{0}^{\infty}f(t)dt+ \sum_{n=1}^{\infty}\int_{0}^{\infty}\sum_{k=0}^{\infty}\frac{f^{(k)}(0) t^{k}(e^{-2 \pi i nt}+e^{2 \pi i nt})}{k!}dt\\
& =   \int_{0}^{\infty}f(t)dt+ \sum_{n=1}^{\infty}\int_{0}^{\infty}f(t)(e^{-2 \pi i nt}+e^{2 \pi i nt})dt.
\end{split}
$$ 
This is enough to write down the formula as 
$$ \sum_{k=1}^{\infty} f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L}  \{ f \} (2 \pi i n).$$

To contrast there is formula of Poisson summation, where $\hat {f}(x)=\int_{-\infty}^{\infty}f (t) e^{-2 \pi i xt}dt$ 


$$\sum_{n=-\infty}^{\infty} f (n)= \sum_{n=-\infty}^{\infty} \hat { f} (n).$$

**Edit:** Proof of Poisson summation from my theorem

Let $x $ be an integer number.

$$\sum_{n=1-x}^{\infty} f (n)=\sum_{n=1}^{\infty} f (n-x)= \sum_{n=-\infty}^{\infty} \int_{0}^{\infty}f(t-x)e^{-2 \pi i nt}dt. $$

For integer $x $ we can write down $\lim\limits_{x \rightarrow \infty}\sum_{n=1-x}^{\infty} f (n)=\lim\limits_{x  \rightarrow \infty}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}f(t-x)e^{-2 \pi i nt}dt=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}f(t)e^{-2 \pi i nt}dt$.

Phase of exponent does not change because  $x $ is integer.