Well, if p is an integer, you should realize that frac1(tau+a)p+1 can be obtained by integrating Q(x)e−2piiax against e−2piitaux where Q(x) is the (unique) polynomial of degree p satisfying Q(m)(0)=e−2piiaQ(m)(1) for m<p and Q(p)=frac(2pii)p+1e−2piia−1.
So, your function is just Q(x)e−2piiax on (0,1) extended by periodicity to the entire line. The polynomial Q can be easily found for each particular p, so if you need some small range of p, you have an exact closed form formula. If you want to consider large p, then it is not so useful but the origianal series gives you a high precision approximation if you keep just the first few terms. Either way, you have an "expression one can work with", don't you?
The thing that totally puzzles me is why you think that your series has any relation to the Hurwitz zeta function.
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