Incomplete Fermi-Dirac integrals and polylogs

Question

The complete Fermi-Dirac integrals $$ F_s(x) = \frac{1}{\Gamma(s+1)} \int\limits_{0}^{\infty} \frac{t^s}{e^{t-x}+1} \: dt $$ are related to the polylogarithms, see http://dlmf.nist.gov/25.12#iii $$ F_s(x) = -\mathrm{Li}_{s+1}(-e^x) $$ Is there any known closed-form relationship of the incomplete Fermi-Dirac integrals $$ F_j(b,x) = \frac{1}{\Gamma(j+1)} \int\limits_{b}^{\infty} \frac{t^j}{e^{t-x}+1} \: dt, \quad b \ge 0 $$ to polylogarithms (at least for integer orders > 1)? For the $j=1$ case I found a formula with Maple $$ F_1(b,x) = \frac{\pi^2}{6} - \frac{b^2}{2} + \frac{x^2}{2} + b \ln(1+ e^{b-x}) + \mathrm{Li}_{2}(-e^{b-x}). $$ For $j \ge 2$ Maple gives a complicated expression including polylogarithms with limits for $t \rightarrow 0$ but no actual closed form. Wolfram Alpha refuses to give answers (it echos the input).

Answer 1

First, several points on polylogarithms:

1) There's a differentiation formula: $$\partial_x Li_s(-e^x)=Li_{s-1}(-e^x).$$

2) the only known exact forms for $Li_s(z)$ are for $s=1,0,-1,\dots$. $$-Li_1(-e^x) = \ln(1+e^x).$$

3) $-Li_s(-1)=\zeta(s)$; $-Li_2(-1) = \frac{\pi^2}{12}$.

In your case, the formula for $F_1$ was obtained via integration by parts (easy to check), the same probably goes for other formulas you've found.