Is there a closed form expression for the following expression:
$$\sum_{j=1}^{N-1} {N-1\choose j} q^j (1-q)^{N-1-j} \frac{c-jd}{e+jd}$$
where $c$, $e$, and $d$ are some real numbers? I wonder if the non-linearity in $j$ doesn't allow for a closed-form expression.
A really closed form expression seems to be difficult (not to say more) but, if you accept gaussian hypergeometric functions $$S_n=\sum_{j=1}^{n-1} {n-1\choose j}\, q^j\, (1-q)^{n-1-j}\, \frac{c-j\,d}{e+j\,d}$$ can write (a CAS told it) $$S_n=\frac{(1-q)^{n-2}}{e (d+e)}T_n$$ where $$T_n=-c (d+e) (1-q)\left(1-(1-q)^{1-n}\right)-$$ $$d (c+e) q(n-1)\,\, _2F_1\left(\frac{d+e}{d},2-n;\frac{2d+e}{d};-\frac{q}{1-q}\right) $$ which seems to be interesting from a computational point of view.