Sums in statistical mechanics I am evaluating the partition function of a system of particles and incurred in sums like
$$
S(a)=\sum_{k=0}^\infty (2k+1)^\frac{\kappa}{2}e^{-(2k+1)a}
$$
being $\kappa\in\mathbb{Z}$ and $a=m_0\beta$ a physical parameter. As usual, $\beta=1/k_BT$. In the low-temperature limit, $a\gg 1$, we are in the lucky situation where the exponential goes rapidly to zero increasing the value of $k$. So, few terms of the series are a good approximation anyway. My question is: Are there techniques to evaluate such sums in closed form as happens for even $\kappa$?
 A: I put a similar sum in Wolfram Alpha (Mathematica) and found:
$$
\sum_{k=0}^m (2 k + 1)^{\kappa/2} e^{-(2 k + 1) a} 
= 2^{\kappa/2} e^{-3 a} \left(e^{2 a} \Phi(e^{-2 a}, -{\kappa/2}, 1/2) - e^{-2 a m} \Phi(e^{-2 a}, -{\kappa/2}, m + 3/2)\right)\;,
$$
where $\Phi$ is the Lerch transcendent. The sum is said to converge for $e^{Re(a)}$ > 1.
Taking the limit as $m \to \infty$:
$$
\sum_{k=0}^\infty (2 k + 1)^{\kappa/2} e^{-(2 k + 1) a} = 2^{\kappa/2} e^{-a} \Phi(e^{-2 a}, -{\kappa/2}, 1/2) 
$$
This seems to be reasonable in the large $a$ limit since
$$
\Phi(e^{-2a},-\kappa/2,1/2) = \sum_{\ell=0}^\infty \frac{e^{-2a\ell}}{(1/2 + \ell)^{-\kappa/2}}
$$
and the overall result in that large $a$ limit is just:
$$
e^{-a}
$$
as might be expected by looking at the $k=0$ term in the sum.

Edit:
It's actually even more straightforward...
$$
\sum_{k=0}^\infty (2 k + 1)^{\kappa/2} e^{-(2 k + 1) a} 
=2^{\kappa/2}\sum_{k=0}^\infty (k + 1/2)^{\kappa/2} e^{-(2 k + 1) a} 
$$
$$
=2^{\kappa/2}\sum_{k=0}^\infty \frac{e^{-a}}{(k+1/2)^{-\kappa/2}}e^{-2 k a}
$$
$$
=2^{\kappa/2}e^{-a}\sum_{k=0}^\infty \frac{(e^{-2 a})^k}{(k+1/2)^{-\kappa/2}}
$$
$$
=2^{\kappa/2}e^{-a}\Phi(e^{-2a}, -\kappa/2, 1/2)\;,
$$
where the last step just identifies the sum with the well-known (or maybe not so well known) Lerch transcendent .
A: Since the second term in the sum is independent of $n$ you can bring it in front. The difficult part of your expression you want to calculate is then
$$ \sum_{n=0}^\infty (2n+1)^{\kappa/2}$$
This can be expressed in terms of the Riemann zeta function modulo some normalization.
