Question Number 28364 by abdo imad last updated on 24/Jan/18
![let give F(x) = (1/(x^2 +1)) prove that ∃ P_n ∈ Z_n [x] / F^((n)) (x)= ((P_n (x))/((1+x^2 )^n )) find a relation of recurence between the P_n .prove that all roots of P_n are reals and smples.](https://www.tinkutara.com/question/Q28364.png)
$${let}\:{give}\:{F}\left({x}\right)\:=\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}\:{prove}\:{that}\:\exists\:{P}_{{n}} \in\:{Z}_{{n}} \left[{x}\right]\:/ \\ $$$${F}^{\left({n}\right)} \left({x}\right)=\:\:\frac{{P}_{{n}} \left({x}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:\:\:{find}\:{a}\:{relation}\:{of}\:{recurence}\:{between}\: \\ $$$${the}\:\:{P}_{{n}} \:.{prove}\:{that}\:{all}\:{roots}\:{of}\:{P}_{{n}} \:{are}\:{reals}\:{and}\:{smples}. \\ $$