User:ViolaMg/sandbox

Roots of n-th function
Let a n-th function $\alpha_n x^n+\alpha_{n-1} x^{n-1}+\ldots+\alpha_0=0 $ for $$\alpha_n\neq 0$$

Dividing in a System the equation like this:

$\begin{cases}

\alpha_n x^n+\alpha_1 x+\alpha_0 =( \alpha_n  -A(\alpha_1-p)^{n} )x^n +p x\\ ( -\alpha_n + A(\alpha_1-p)^{n})x^{n} - p x =\alpha_{n-1}x^{n-1}+\ldots+\alpha_{2} x^2

\end{cases}(*)$

Solve first with generalized Hypergeometric Function into variable x

$$x= - \frac{\alpha_0}{\alpha_1- p}\,_{n-1}F_{n-2}\left(\frac{1}{n},\frac{2}{n},\ldots,\frac{n-1}{n};\frac{n}{n-1},\frac{n-2}{n-1},\frac{n-3}{n-1},\ldots,\frac{2}{n-1}; \frac{ (-1)^n A(\alpha_1 -  p)^n*(\alpha_0)^{n-1}*n^n}{( \alpha_1-p)^n (n-1)^{n-1}}\right)$$

For $$\begin{align} & (-1)^{n-1}   A\alpha_0^{n-1}\,_{n-1}F_{n-2}^{n-1}=1\\ & case \, n=2k+1 \begin{cases}

A^\frac{1}{n-1} \alpha_0=\frac{- 1}{F}\\

A^{\frac{n}{n-1}-1}\alpha_0 + 2=0 \end{cases}\\ & case \, n=2k  \begin{cases} \frac{ 1}{A^\frac{1}{n-1}\alpha_0}= - F\\ A^{\frac{-n}{n-1}+1} \alpha_0^{-n+n-1}+A^\frac{-1}{n-1} \alpha_0^{-1} +1 = 0 \end{cases} \end{align}$$

and solve second of $$(*)$$

Example

R: $$x = \,_4F_3\left(\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ;\frac{1}{2}, \frac{3}{4}, \frac{5}{4} ;\frac{3125}{256}\right)$$
 * $$x^5-x+1=0$$


 * $$x^6-2x^4+2x^3-6x^2+6x-12=0 $$


 * $$x^7+2x^3-2x+ 3=0 $$