I am trying to prove, for the general case whereby $\zeta(\cdot\,,\cdot)$ is the Hurwitz-Zeta function, and $a\in \mathbb{N}$, that $$\mathcal{L} = \lim\limits_{n\to\infty^{+}}\sum\limits_{i=1}^{n}\left(\frac{n}{n+1-i}\right)^{a}\Big[\zeta(a,i)-\zeta(a,n+1)\Big] = 2\zeta(a-1),\quad (a>2)$$
Importantly, I was hoping to accomplish this without transitioning the limit of the sum to a Riemann integral.
However, evaluating the limit is tricky because of the behavior of the factor $f_n=[n/(n+1-i)]$ since $i$ spans all the way to $n$ itself. I realized after a bit that $$\mathcal{L} = 2\zeta(a-1)=2\sum\limits_{i\geq 1}\sum\limits_{k\geq 0}\left(\frac{1}{k+i} \right)^a=2\sum\limits_{i\geq 1}\zeta(a,i)$$
where the summand corresponds to the main $i$-dependent function in the original sum.
Although I still don't know how to prove the limit rigorously, it made me wonder if it's true more generally. For example, let $g(a,i)=\exp(-ai)$ and $h(a,i)=(1/i!)^a$ then is it true that for some constant $c$
$$\lim\limits_{n\to\infty^{+}}\sum\limits_{i=1}^{n}\left(\frac{n}{n+1-i}\right)^{a}g(a,i)\stackrel{?}{=}c\sum\limits_{i\geq 1}g(a,i),\quad (a>0)$$
$$\lim\limits_{n\to\infty^{+}}\sum\limits_{i=1}^{n}\left(\frac{n}{n+1-i}\right)^{a}h(a,i)\stackrel{?}{=}c\sum\limits_{i\geq 1}h(a,i),\quad (a>0)$$
Numerical evaluations show that $c$ is most likely $1$ in both of these cases. Assuming $\lim\limits{n\to\infty} f_n \sim 1$, then the limits with $g$ and $h$ make sense. However why doesn't the first limit I proposed follow the same convention and instead have $c=2$? It appears that for $0<k<a$, factor $c$ is reduced back to $1$ (at least by numerical evaluations).
$$\lim\limits_{n\to\infty^{+}}\sum\limits_{i=1}^{n}\left(\frac{n}{n+1-i}\right)^{k}\Big[\zeta(a,i)-\zeta(a,n+1)\Big] \stackrel{?}{=} \zeta(a-1),\quad (a>2)$$
I'm wondering how you guys would approach any of these limits, since I don't have a convincing proof for any of them. Potentially would also like to know how/why the exponent on $f_n$ is able to change things in the first limit example.
Please share your thoughts!