This is my first post in stackoverflow so please forbear my inexperience in setting up and formatting the question.

I'm working on a special algorithm to implement a more accurate effective mass calculation for hole carriers in Silicon into mathematica. This rather involved algorithm uses incomplete Fermi-Dirac Integrals of the form:

Fincomplete[j_, x_, b_] :=1/Gamma[j + 1] Integrate[t^j/(E^(t - x) + 1), {t, b,\[Infinity]}, Assumptions -> j > 0]

A call to this function with b=0 yields the complete Fermi-Dirac integral representation by the nicely implemented PolyLog Function within mathematica.

In[1]:=  Fincomplete[j,x,0]
Out[1]:= -PolyLog[1 + j, -E^x]

However if one wants to evaluate the incomplete Fermi-Dirac integral the expression stays unevaluated, indicating that there is no straightforward solution available.

In[2]:=  Fincomplete[j,x,b]
Out[2]:= Integrate[t^j/(1 + E^(t - x)), {t, b, \[Infinity]}, Assumptions -> j > 0]/Gamma[1 + j]

Now my question(s):

1. Any idea which of the multitude of mathematical functions within mathematica could be leveraged to implement the incomplete Fermi-Dirac integral? Could the Nielsen generalized polylogarithm function mentioned within the mathematica help system be of any service?
2. Does anybody have already this integral implemented? There are some series expansions quoted on the web Computation of the complete and incomplete Fermi-Dirac integral. However the reference quotes that the series expansions are converging rather slowly and I would like to leave the implementation of above reference as last resort, since this would take some time and maybe there is an easier way in mathematica?

Thanks in advance for all of your help!