Solution to generalized-polynomial equation? Is it possible to obtain the solution to this generalized-polynomial equation?
$$b x^a - x +c =0$$
with $-1<a<1$, $b>0$, $c>0$ and $x>0$.

In general, only by numerical methods or series.  The following series solution in powers of $b c^{a-1}$ converges if $b c^{a-1}$ is small :
$$ \eqalign{x &= c + b c^a + \frac{ca}{b} (b c^{a-1})^2 + \frac{ca(3a-1)}{2} (b c^{a-1})^3 + \ldots\cr
&= c + b c^a + c \sum_{k=2}^\infty \frac{(b c^{a-1})^k}{k!} \prod_{j=0}^{k-2} (ka-j)}$$
