I was going through some old stuff when I happened upon fractional summations. They come of the form $\sum_{k=a}^bf(k)$ for $b-a\notin\mathbb Z$. One such example is the extended harmonic numbers, which has:
$$\sum_{k=1}^n\frac1k=\sum_{k=1}^\infty\frac1k-\frac1{k+n}$$
Which got me thinking. If we had $f(x)$ be continuous and $\lim\limits_{x\to\infty}f(x)=0$, then one might perchance define fractional summations as follows:
$$\sum_{k=1}^xf(k)\equiv\sum_{k=1}^\infty f(k)-f(k+x)\forall x\notin\mathbb Z$$
For example,
$$\sum_{k=1}^x\frac1{k^s}=\sum_{k=1}^\infty\frac1{k^s}-\frac1{(k+x)^s}=\zeta(s)-\zeta(s,x+1),s\ne1,s>0$$
This got me thinking:
What other ways can we extend summations to fractional orders?
Anything to worry about the above definition?
Where would these fractional summations appear?
Related: Indefinite summation