Hello infinity series enjoyers. If my calulations are correct, I think I made some integral representation of any divergent and convergent seris of any order s. All calulations and the question are down below.
$\displaystyle \sum\limits^s_{k=x} \;\!\!\;\! \!\!\!\!\!\!\!\!\! \lower -0.2pt {\infty} \quad \!\!\! f (k) =\overbrace { \sum_{k_{s-1}=x} ^{\infty} ... \sum_{k_1=k_2}^{\infty} \sum_{k_0=k_1}^{\infty}}^{s}f (k_0) =\sum_{k=x}^{\infty}f (k)\frac {( k-x+s-1)!}{\Gamma (s)(k-x)!}$
We can prove that (Infinite series as integral representation)
$\displaystyle \sum\limits^1_{k=x} \;\!\!\;\! \!\!\!\!\!\!\!\!\! \lower -0.2pt {\infty} \quad \!\!\! f (k) = \lim\limits_{h \rightarrow 0} \sum_{n=0}^{\infty}\frac {\frac{d^{n}}{dx^n}F^s (x) \frac{d^{n}}{dh^n} \left( \frac{-h}{e^h-1} \right) }{n!} =-\frac {i \pi}{2}\int_{-\frac {1}{2}-i\infty}^{-\frac {1}{2}+i\infty} \csc^2 (\pi t) \int f(x+t)dt dt $
Now we have to just substitute first equation to secound
$\displaystyle \sum\limits^s_{k=x} \;\!\!\;\! \!\!\!\!\!\!\!\!\! \lower -0.2pt {\infty} \quad \!\!\! f (k) =-\frac {i \pi }{2}\int_{-\frac {1}{2}-i\infty}^{-\frac {1}{2}+i\infty} csc^2 (\pi t) \int f (t+x)\frac {( t+s-1)!}{\Gamma (s)(t)!} dt dt =\frac {i \pi (-1)^{s}}{2\Gamma (s)}\int_{-\frac {1}{2}-i\infty}^{-\frac {1}{2}+i\infty} F^{s}(x+t)\Psi_s (t)dt$
Where $F^s$ is s ordered antideritative of f and $\Psi_s(t)$ is defined down below.
$\displaystyle \Psi_s(t)= \sum_{n=0} ^{\infty} \left [ \frac {d^{n}}{dt^{n}}csc^2 (\pi t) \right] \left [ \frac {d^{s-1-n}}{dt^{s-1-n}} \frac {(t+s-1)!}{ t!} {s\choose n+1} \right]$
But we can write $F^{s}(x+t)$ as Taylor series and compere it to generalised Euler-Maclouren Summation to determinate $(-1)^{s}$ for non itintegers
$\displaystyle \sum\limits^s_{k=x} \;\!\!\;\! \!\!\!\!\!\!\!\!\! \lower -0.2pt {\infty} \quad \!\!\! f (k) = \sum_{n=0}^{\infty}\frac { F^{s-n}(x)}{n!} (-1)^s\int_{-\frac {1}{2}-i\infty}^{-\frac {1}{2}+i\infty}(-t)^n \frac {i \pi }{2\Gamma (s)}\Psi_s (t)dt = \lim\limits_{h \rightarrow 0} \sum_{n=0}^{\infty} \frac {F^{s-n} (x) \frac{d^{n}}{dh^n} \left( \frac{-h}{e^h-1} \right)^s }{n!} $
And that imply
$\displaystyle \sum\limits^s_{k=x} \;\!\!\;\! \!\!\!\!\!\!\!\!\! \lower -0.2pt {\infty} \quad \!\!\! f (k) =\frac {i \pi e^{i \pi s}}{2\Gamma (s)}\int_{-\frac {1}{2}-i\infty}^{-\frac {1}{2}+i\infty} F^{s}(x+t)\Psi_s (t)dt $
Is there any simple equation for $\Psi_s (t) $? Maybe I overcomplicated some transformations.
\sum\!\!\!\!\!\!\!\!\!\inftygives $\sum!!!!!!!!!\infty$ or you can say $f^s(x)=\underbrace{f(f(\dots f(x)))}_{s \text{ number of } f\text{‘s}}$ – Тyma Gaidash Nov 28 '22 at 18:15