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(Please pardon the use of nonstandard terminology, as I know not the accepted names for my entities of interest.)

Some personal research I have been doing led me to consider series of the form $$\sum_{k=1}^\infty\frac1{(k^2+bk+c)^s}$$

for complex $s$ and real $b,c$. I have been provisionally calling them "quadratic Dirichlet series", in the sense that a quadratic polynomial appears in the denominator of the $k$-th term.

I want to ask if these have ever been previously studied in the literature, and if there are more established names for them. I am particularly interested in references regarding their (symbolic or numeric) evaluation, but any related paper would be appreciated.

An obvious easy case is when $b=c=0$, where it degenerates to the usual $\zeta$ function. Apart from that, I haven't been able to find any simpler closed forms for these sums.

The Epstein function seemed close, but the expressions in the denominator are multi-index quadratic forms as opposed to the simpler(?) case I have.


Bonus: if there are any studies on the "polylogarithmic" generalization $$\sum_{k=1}^\infty\frac{z^k}{(k^2+bk+c)^s}$$ I would like to see them too.

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  • $\begingroup$ For integer $s$, closed form appears to involve polygamma. $\endgroup$
    – joro
    Apr 3, 2016 at 7:01
  • $\begingroup$ I propose the term: Rankin-Selberg convolution of Hurwitz zeta functions. $\endgroup$
    – user1688
    Apr 3, 2016 at 7:23
  • $\begingroup$ it could be possible to locate its singularities by expanding $f(x) = \sum_{k=1}^\infty e^{-x(k^2+bk+c)}$ at $x=0$ since $\Gamma(s) \sum_{k=1}^\infty (k^2+bk+c)^{-s} = \int_0^\infty x^{s-1} f(x) dx$ . $\endgroup$
    – reuns
    Apr 3, 2016 at 8:16
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    $\begingroup$ @joro, makes sense; any infinite sum with rational function terms is ultimately expressible in terms of generalized harmonic numbers (equivalently, polygamma functions). It makes you wonder if these functions can be analytically continued to $s$ a nonpositive integer, tho. $\endgroup$ Apr 3, 2016 at 14:10

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