Can we sum a series over noninteger bounds? Are there any ways to generalize series to noninteger bounds? For example, is there any way to make sense of
$$
f(\nu)=\sum_{k=\nu}^\infty a_k
$$
for $\nu\in\Bbb C$? I would expect that if such a generalization exists, it should obey have the same properties as the case where $\nu$ is an integer.  In particular, I would expect it to obey the recurrence relation
$$
f(\nu)-f(\nu+1)=a_\nu.
$$
We find in chapter 2 Sums of  Concrete Mathematics
by R.L. Graham, D.E.  Knuth  and O.  Patashnik the following statement (see formula 2.4):

Formally, we write
\begin{align*}
\sum_{P(k)} a_k\tag{2.4}
\end{align*}
as an abbreviation for the sum of all terms $a_k$ such that $k$  is an integer satisfying a given property $P(k)$. (A property $P(k)$ is any statement about $k$ that can be either true or false.)

We see the index $k$ is an integer. Sometimes we have the situation that the upper limit of a sum is not an integer. In this situation we take the greatest integer less or equal to the upper limit (floor function).
\begin{align*}
\sum_{k=0}^{\frac{n}{2}} a_k=\sum_{k=0}^{\color{blue}{\left\lfloor\frac{n}{2}\right\rfloor}} a_k=a_0+a_1+\cdots+a_{\left\lfloor\frac{n}{2}\right\rfloor}
\end{align*}
Similarly the lower limit is sometimes not an integer. In this situation we take the smallest integer greater or equal to the lower limit
(ceiling function).
\begin{align*}
\sum_{k=\frac{m}{2}}^{\frac{n}{2}} a_k=\sum_{k=\color{blue}{\left\lceil\frac{m}{2}\right\rceil}}^{\left\lfloor\frac{n}{2}\right\rfloor} a_k=a_{\left\lceil\frac{m}{2}\right\rceil}+a_{\left\lceil\frac{m}{2}\right\rceil+1}+\cdots+a_{\left\lfloor\frac{n}{2}\right\rfloor}
\end{align*}

If the lower limit of the index is equal to $e$, we have consequently
\begin{align*}
\sum_{k=e}^\infty a_k=\sum_{k=\left\lceil e\right\rceil}^\infty a_k=\sum_{\color{blue}{k=3}}^\infty a_k
\end{align*}
I guess you could write something like
$$
\sum_{n = 0}^\infty f(e + n)
$$