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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.

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  • $\begingroup$ Is $(c - jd)/(e+jd)$ always positive for all $j$? What about the magnitudes? $\endgroup$ – Lee David Chung Lin Apr 25 at 1:36
  • $\begingroup$ The fraction $(c-jd)/(e+jd)$ is not necessarily positive for all $j$. Would it help if it was always positive? $\endgroup$ – Econquer Apr 25 at 1:50
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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 $$T_n=-c (d+e) (1-q)\left(1-(1-q)^{1-n}\right)-d (c+e) q\,\, _2F_1\left(\frac{d+e}{d},2-n;\frac{2d+e}{d};-\frac{q}{1-q}\right) (n-1)$$ which seems to be interesting from a computational point of view.

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