mathematical physics - Combinatorial sum in a problem with a Fermi gas

I'm solving a problem involving a Fermi gas. There is a specific sum I cannot figure my way around.

A set of equidistant levels, indexed by $m=0,1,2 \ldots$, is populated by spinless fermions with population numbers $\nu_m =0$ or $1$. I need to compute the following sum over the set of all possible configurations $\{ \nu_l \}$:

$Q(\beta,\beta_c) = \sum_{\{ \nu_l \}} \sum_{l} \prod_m \exp({\beta_c \, l \, \nu_l}-{ [ \beta \, m + i \phi] \, \nu_m} )$.

Any hints on how to deal with this are appreciated. This is not homework, it is a research problem.

It is known that $\beta >0$, $\beta_c>0$, and $\phi \in [0; 2 \pi ]$.

EDIT: corrected with the complex phase (the sum is coming from a generating function)

Answer

This is not an answer just some thoughts from playing with the expression. I've read the question before you included the phase, so for now let $\phi = 0$ (sorry it this makes my response useless for you).

I'll simply write $Z$ instead of your $Q(\beta, \beta_c)$ and also drop the arguments where obvious. Denote by $Z_{abc\dots}$ the partition function where we do not include the sites at $a, b, c, \dots$ in the problem. Also denote $f_k = 1 + \exp(-\beta k)$ and $g_k = 1 + \exp((\beta_c - \beta) k)$.

Now (unless I screwed up), by summing over the site at $k$ we can get the relation

$$Z = f_kZ_k + g_k \sum_{\nu \setminus k} \prod_{m \neq k} \exp(-\beta m \nu_m)$$ and iterating it $$Z = \left( \prod_{m \in abc\dots z} f_m \right) Z_{abc\dots z} +$$ $$\left(g_a f_b \dots f_z + f_a g_b \dots f_z + \cdots + f_a f_b \dots g_z) \right) \sum_{\nu \setminus abc\dots z} \prod_{m \neq abc \dots z} \exp(-\beta m \nu_m).$$

It is a simple observation that for the reduces system consisting of a single level $a$ we get $Z_{bc \ldots z} = g_a$ so the first term above gives a similar contribution like the other terms (all but one factors are $f$ and one of them is $g$). Therefore, we can write

$$Z = \left( \prod_{m} f_m \right) \left ( \sum_k \frac{g_k} { f_k} \right).$$

These expressions are exact in case we have finite number of states. Otherwise they are just formal and are to be understood as limits only if everything converges.